% Mizar ND problem: t1_bhsp_7,bhsp_7,88,57 fof(dh_c1_4__bhsp_7,definition, ( ( ( ~ v3_struct_0(c1_4__bhsp_7) & v3_rlvect_1(c1_4__bhsp_7) & v4_rlvect_1(c1_4__bhsp_7) & v5_rlvect_1(c1_4__bhsp_7) & v6_rlvect_1(c1_4__bhsp_7) & v7_rlvect_1(c1_4__bhsp_7) & v2_bhsp_1(c1_4__bhsp_7) & l1_bhsp_1(c1_4__bhsp_7) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) & m2_relset_1(B,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) ) => ( r1_bhsp_6(c1_4__bhsp_7,A,B) <=> ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,A) & ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(E) & r1_tarski(E,A) & r1_xboole_0(D,E) & r1_xreal_0(C,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,E,B))) ) ) ) ) ) ) ) ) ) ) => ! [F] : ( ( ~ v3_struct_0(F) & v3_rlvect_1(F) & v4_rlvect_1(F) & v5_rlvect_1(F) & v6_rlvect_1(F) & v7_rlvect_1(F) & v2_bhsp_1(F) & l1_bhsp_1(F) ) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(F))) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(F),k6_supinf_1) & m2_relset_1(H,u1_struct_0(F),k6_supinf_1) ) => ( r1_bhsp_6(F,G,H) <=> ! [I] : ( m1_subset_1(I,k1_numbers) => ~ ( ~ r1_xreal_0(I,0) & ! [J] : ( ( v1_finset_1(J) & m1_subset_1(J,k1_zfmisc_1(u1_struct_0(F))) ) => ~ ( ~ v1_xboole_0(J) & r1_tarski(J,G) & ! [K] : ( ( v1_finset_1(K) & m1_subset_1(K,k1_zfmisc_1(u1_struct_0(F))) ) => ~ ( ~ v1_xboole_0(K) & r1_tarski(K,G) & r1_xboole_0(J,K) & r1_xreal_0(I,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(F),k33_binop_2,K,H))) ) ) ) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4__bhsp_7),file(bhsp_7,c1_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,c1_4__bhsp_7)]). fof(dh_c2_4__bhsp_7,definition, ( ( m1_subset_1(c2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) & m2_relset_1(A,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) ) => ( r1_bhsp_6(c1_4__bhsp_7,c2_4__bhsp_7,A) <=> ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ r1_xreal_0(B,0) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(C,D) & r1_xreal_0(B,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,A))) ) ) ) ) ) ) ) ) ) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) & m2_relset_1(F,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) ) => ( r1_bhsp_6(c1_4__bhsp_7,E,F) <=> ! [G] : ( m1_subset_1(G,k1_numbers) => ~ ( ~ r1_xreal_0(G,0) & ! [H] : ( ( v1_finset_1(H) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(H) & r1_tarski(H,E) & ! [I] : ( ( v1_finset_1(I) & m1_subset_1(I,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(I) & r1_tarski(I,E) & r1_xboole_0(H,I) & r1_xreal_0(G,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,I,F))) ) ) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_4__bhsp_7),file(bhsp_7,c2_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,c2_4__bhsp_7)]). fof(dh_c3_4__bhsp_7,definition, ( ( ( v1_funct_1(c3_4__bhsp_7) & v1_funct_2(c3_4__bhsp_7,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) & m2_relset_1(c3_4__bhsp_7,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) ) => ( r1_bhsp_6(c1_4__bhsp_7,c2_4__bhsp_7,c3_4__bhsp_7) <=> ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(B,C) & r1_xreal_0(A,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) & m2_relset_1(D,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) ) => ( r1_bhsp_6(c1_4__bhsp_7,c2_4__bhsp_7,D) <=> ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( ~ r1_xreal_0(E,0) & ! [F] : ( ( v1_finset_1(F) & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(F) & r1_tarski(F,c2_4__bhsp_7) & ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(G) & r1_tarski(G,c2_4__bhsp_7) & r1_xboole_0(F,G) & r1_xreal_0(E,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,G,D))) ) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c3_4__bhsp_7),file(bhsp_7,c3_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,c3_4__bhsp_7)]). fof(e1_4_1__bhsp_7,assumption,( r1_bhsp_6(c1_4__bhsp_7,c2_4__bhsp_7,c3_4__bhsp_7) ), introduced(assumption,[file(bhsp_7,e1_4_1__bhsp_7)]), [interesting(0.65),axiom,file(bhsp_7,e1_4_1__bhsp_7)]). fof(existence_l2_struct_0,axiom,( ? [A] : l2_struct_0(A) ), file(struct_0,l2_struct_0), [interesting(0.9),axiom,file(struct_0,l2_struct_0)]). fof(dt_l2_struct_0,axiom,( ! [A] : ( l2_struct_0(A) => l1_struct_0(A) ) ), file(struct_0,l2_struct_0), [interesting(0.9),axiom,file(struct_0,l2_struct_0)]). fof(rc4_struct_0,theorem,( ? [A] : ( l2_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc4_struct_0), [interesting(0.9),axiom,file(struct_0,rc4_struct_0)]). fof(existence_l1_rlvect_1,axiom,( ? [A] : l1_rlvect_1(A) ), file(rlvect_1,l1_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l1_rlvect_1)]). fof(dt_l1_rlvect_1,axiom,( ! [A] : ( l1_rlvect_1(A) => l2_struct_0(A) ) ), file(rlvect_1,l1_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l1_rlvect_1)]). fof(rc2_finset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), [interesting(0.9),axiom,file(finset_1,rc2_finset_1)]). fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(existence_l2_rlvect_1,axiom,( ? [A] : l2_rlvect_1(A) ), file(rlvect_1,l2_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l2_rlvect_1)]). fof(dt_k3_supinf_1,axiom,( $true ), file(supinf_1,k3_supinf_1), [interesting(0.9),axiom,file(supinf_1,k3_supinf_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_l2_rlvect_1,axiom,( ! [A] : ( l2_rlvect_1(A) => l1_rlvect_1(A) ) ), file(rlvect_1,l2_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,l2_rlvect_1)]). fof(cc1_funct_2,theorem,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_partfun1(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) ) ) ) ), file(funct_2,cc1_funct_2), [interesting(0.9),axiom,file(funct_2,cc1_funct_2)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(fc1_ordinal2,theorem, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]). fof(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc1_funct_2,theorem,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,A,B) ) ), file(funct_2,rc1_funct_2), [interesting(0.9),axiom,file(funct_2,rc1_funct_2)]). fof(rc2_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(projectivity_k16_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), [interesting(0.9),axiom,file(complex1,k16_complex1)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(existence_l1_bhsp_1,axiom,( ? [A] : l1_bhsp_1(A) ), file(bhsp_1,l1_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,l1_bhsp_1)]). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_k6_supinf_1,definition,( k6_supinf_1 = k1_numbers ), file(supinf_1,k6_supinf_1), [interesting(0.9),axiom,file(supinf_1,k6_supinf_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_complex1), [interesting(0.9),axiom,file(complex1,k16_complex1)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_k6_supinf_1,axiom, ( ~ v1_xboole_0(k6_supinf_1) & m1_subset_1(k6_supinf_1,k1_zfmisc_1(k3_supinf_1)) ), file(supinf_1,k6_supinf_1), [interesting(0.9),axiom,file(supinf_1,k6_supinf_1)]). fof(dt_l1_bhsp_1,axiom,( ! [A] : ( l1_bhsp_1(A) => l2_rlvect_1(A) ) ), file(bhsp_1,l1_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,l1_bhsp_1)]). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc1_relset_1,theorem,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc5_funct_2,theorem,( ! [A,B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc5_funct_2), [interesting(0.9),axiom,file(funct_2,cc5_funct_2)]). fof(cc6_funct_2,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & ~ v1_xboole_0(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc6_funct_2), [interesting(0.9),axiom,file(funct_2,cc6_funct_2)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc14_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]). fof(fc1_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]). fof(fc1_xboole_0,theorem,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc3_struct_0,theorem,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]). fof(rc5_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(projectivity_k18_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ), file(complex1,k18_complex1), [interesting(0.9),axiom,file(complex1,k18_complex1)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(symmetry_r1_xboole_0,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), [interesting(0.9),axiom,file(xboole_0,r1_xboole_0)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(redefinition_k18_complex1,definition,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(A) = k16_complex1(A) ) ), file(complex1,k18_complex1), [interesting(0.9),axiom,file(complex1,k18_complex1)]). fof(dt_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k18_complex1(A),k1_numbers) ) ), file(complex1,k18_complex1), [interesting(0.9),axiom,file(complex1,k18_complex1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k33_binop_2,axiom, ( v1_funct_1(k33_binop_2) & v1_funct_2(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ), file(binop_2,k33_binop_2), [interesting(0.9),axiom,file(binop_2,k33_binop_2)]). fof(dt_k5_bhsp_5,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) & v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(B)) & v1_funct_1(E) & v1_funct_2(E,B,A) & m1_relset_1(E,B,A) ) => m1_subset_1(k5_bhsp_5(A,B,C,D,E),A) ) ), file(bhsp_5,k5_bhsp_5), [interesting(0.9),axiom,file(bhsp_5,k5_bhsp_5)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(dt_c1_4__bhsp_7,assumption, ( ~ v3_struct_0(c1_4__bhsp_7) & v3_rlvect_1(c1_4__bhsp_7) & v4_rlvect_1(c1_4__bhsp_7) & v5_rlvect_1(c1_4__bhsp_7) & v6_rlvect_1(c1_4__bhsp_7) & v7_rlvect_1(c1_4__bhsp_7) & v2_bhsp_1(c1_4__bhsp_7) & l1_bhsp_1(c1_4__bhsp_7) ), introduced(assumption,[file(bhsp_7,c1_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,c1_4__bhsp_7)]). fof(dt_c2_4__bhsp_7,assumption,( m1_subset_1(c2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), introduced(assumption,[file(bhsp_7,c2_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,c2_4__bhsp_7)]). fof(dt_c3_4__bhsp_7,assumption, ( v1_funct_1(c3_4__bhsp_7) & v1_funct_2(c3_4__bhsp_7,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) & m2_relset_1(c3_4__bhsp_7,u1_struct_0(c1_4__bhsp_7),k6_supinf_1) ), introduced(assumption,[file(bhsp_7,c3_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,c3_4__bhsp_7)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc1_finset_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]). fof(cc1_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(cc2_finset_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(fc12_binop_2,theorem, ( v1_funct_1(k33_binop_2) & v1_funct_2(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & v1_binop_1(k33_binop_2,k1_numbers) & v2_binop_1(k33_binop_2,k1_numbers) & v1_setwiseo(k33_binop_2,k1_numbers) ), file(binop_2,fc12_binop_2), [interesting(0.9),axiom,file(binop_2,fc12_binop_2)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(fc3_binop_2,theorem, ( v1_funct_1(k33_binop_2) & v1_funct_2(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & v1_binop_1(k33_binop_2,k1_numbers) & v2_binop_1(k33_binop_2,k1_numbers) ), file(binop_2,fc3_binop_2), [interesting(0.9),axiom,file(binop_2,fc3_binop_2)]). fof(rc1_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]). fof(rc1_xboole_0,theorem,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]). fof(rc2_xboole_0,theorem,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]). fof(rc3_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]). fof(rc4_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]). fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(dh_c1_4_1_1__bhsp_7,definition, ( ( m1_subset_1(c1_4_1_1__bhsp_7,k1_numbers) => ~ ( ~ r1_xreal_0(c1_4_1_1__bhsp_7,0) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ) ) ) => ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(E) & r1_tarski(E,c2_4__bhsp_7) & r1_xboole_0(D,E) & r1_xreal_0(C,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,E,c3_4__bhsp_7))) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_1_1__bhsp_7),file(bhsp_7,c1_4_1_1__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_1_1__bhsp_7)]). fof(e1_4_1_1__bhsp_7,assumption,( ~ r1_xreal_0(c1_4_1_1__bhsp_7,0) ), introduced(assumption,[file(bhsp_7,e1_4_1_1__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,e1_4_1_1__bhsp_7)]). fof(fc15_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc15_xreal_0)]). fof(fc16_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc16_xreal_0)]). fof(fc19_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc19_xreal_0)]). fof(fc20_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc20_xreal_0)]). fof(fc21_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc21_xreal_0)]). fof(fc22_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc22_xreal_0)]). fof(fc24_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc24_xreal_0)]). fof(fc27_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc27_xreal_0)]). fof(fc28_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc28_xreal_0)]). fof(fc29_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc29_xreal_0)]). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(fc17_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc17_xreal_0)]). fof(fc18_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc18_xreal_0)]). fof(fc23_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc23_xreal_0)]). fof(fc30_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc30_xreal_0)]). fof(fc4_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc4_xreal_0)]). fof(fc5_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc5_xreal_0)]). fof(fc6_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc6_xreal_0)]). fof(rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,theorem,( k6_xcmplx_0(1,k7_xcmplx_0(1,2)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2)]). fof(rqRealDiff__k6_xcmplx_0__r2_r1_r1,theorem,( k6_xcmplx_0(2,1) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1)]). fof(spc4_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(A,k7_xcmplx_0(B,C)) = k7_xcmplx_0(k3_xcmplx_0(A,B),C) ) ), file(arithm,spc4_arithm), [interesting(0.9),axiom,file(arithm,spc4_arithm)]). fof(spc7_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(A,k3_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(t3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), [interesting(0.9),axiom,file(arithm,t3_arithm)]). fof(t6_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), [interesting(0.9),axiom,file(arithm,t6_arithm)]). fof(commutativity_k3_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(redefinition_k5_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), [interesting(0.9),axiom,file(real_1,k5_real_1)]). fof(redefinition_k6_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), [interesting(0.9),axiom,file(real_1,k6_real_1)]). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), [interesting(0.9),axiom,file(real_1,k5_real_1)]). fof(dt_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), [interesting(0.9),axiom,file(real_1,k6_real_1)]). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k7_xcmplx_0)]). fof(dh_c1_4_1__bhsp_7,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ r1_xreal_0(B,0) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(C,D) & r1_tarski(D,c2_4__bhsp_7) & r1_xreal_0(B,k18_complex1(k5_real_1(A,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7)))) ) ) ) ) ) ) ) => ( m1_subset_1(c1_4_1__bhsp_7,k1_numbers) & ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( ~ r1_xreal_0(E,0) & ! [F] : ( ( v1_finset_1(F) & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(F) & r1_tarski(F,c2_4__bhsp_7) & ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(F,G) & r1_tarski(G,c2_4__bhsp_7) & r1_xreal_0(E,k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,G,c3_4__bhsp_7)))) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_1__bhsp_7),file(bhsp_7,c1_4_1__bhsp_7)]), [interesting(0.65),axiom,file(bhsp_7,c1_4_1__bhsp_7)]). fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,( k6_xcmplx_0(0,0) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(redefinition_k10_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k10_binop_2(A,B) = k6_xcmplx_0(A,B) ) ), file(binop_2,k10_binop_2), [interesting(0.9),axiom,file(binop_2,k10_binop_2)]). fof(dt_k10_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k10_binop_2(A,B),k1_numbers) ) ), file(binop_2,k10_binop_2), [interesting(0.9),axiom,file(binop_2,k10_binop_2)]). fof(d6_bhsp_6,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k6_supinf_1) & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) ) => ( r1_bhsp_6(A,B,C) <=> ? [D] : ( m1_subset_1(D,k1_numbers) & ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( ~ r1_xreal_0(E,0) & ! [F] : ( ( v1_finset_1(F) & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(F) & r1_tarski(F,B) & ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( r1_tarski(F,G) & r1_tarski(G,B) & r1_xreal_0(E,k18_complex1(k10_binop_2(D,k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C)))) ) ) ) ) ) ) ) ) ) ) ) ), file(bhsp_6,d6_bhsp_6), [interesting(0.9),axiom,file(bhsp_6,d6_bhsp_6)]). fof(e2_4_1__bhsp_7,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ r1_xreal_0(B,0) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(C,D) & r1_tarski(D,c2_4__bhsp_7) & r1_xreal_0(B,k18_complex1(k5_real_1(A,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7)))) ) ) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc2_finset_1,rc3_funct_1,rc4_struct_0,existence_l1_rlvect_1,dt_k5_ordinal2,dt_l1_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc20_xreal_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k3_supinf_1,dt_k5_numbers,dt_k6_xcmplx_0,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xboole_0,fc5_xreal_0,fc6_membered,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,t1_numerals,t1_real,t1_subset,t4_arithm,t4_real,t4_subset,t5_subset,t8_boole,projectivity_k18_complex1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_k10_binop_2,redefinition_k18_complex1,redefinition_k5_real_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k10_binop_2,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc0_boole,spc0_numerals,e1_4_1__bhsp_7,d6_bhsp_6]), [interesting(0.65),file(bhsp_7,e2_4_1__bhsp_7),[file(bhsp_7,e2_4_1__bhsp_7)]]). fof(dt_c1_4_1__bhsp_7,plain,( m1_subset_1(c1_4_1__bhsp_7,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[dh_c1_4_1__bhsp_7,e2_4_1__bhsp_7]), [interesting(0.65),file(bhsp_7,c1_4_1__bhsp_7),[file(bhsp_7,c1_4_1__bhsp_7)]]). fof(dt_c1_4_1_1__bhsp_7,assumption,( m1_subset_1(c1_4_1_1__bhsp_7,k1_numbers) ), introduced(assumption,[file(bhsp_7,c1_4_1_1__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_1_1__bhsp_7)]). fof(dh_c2_4_1_1__bhsp_7,definition, ( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(A,B) & r1_tarski(B,c2_4__bhsp_7) & r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7)))) ) ) ) => ( v1_finset_1(c2_4_1_1__bhsp_7) & m1_subset_1(c2_4_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(c2_4_1_1__bhsp_7) & r1_tarski(c2_4_1_1__bhsp_7,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(c2_4_1_1__bhsp_7,C) & r1_tarski(C,c2_4__bhsp_7) & r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7)))) ) ) ) ), introduced(definition,[new_symbol(c2_4_1_1__bhsp_7),file(bhsp_7,c2_4_1_1__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c2_4_1_1__bhsp_7)]). fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,( k6_xcmplx_0(1,0) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealDiff__k6_xcmplx_0__r2_r0_r2,theorem,( k6_xcmplx_0(2,0) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2)]). fof(rqRealDiff__k6_xcmplx_0__r2_r2_r0,theorem,( k6_xcmplx_0(2,2) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0)]). fof(rqRealDiff__k6_xcmplx_0__rn1d2_r0_rn1d2,theorem,( k6_xcmplx_0(k7_xcmplx_0(1,2),0) = k7_xcmplx_0(1,2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_r0_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_r0_rn1d2)]). fof(rqRealDiff__k6_xcmplx_0__rn1d2_rn1d2_r0,theorem,( k6_xcmplx_0(k7_xcmplx_0(1,2),k7_xcmplx_0(1,2)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_rn1d2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_rn1d2_r0)]). fof(t2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), [interesting(0.9),axiom,file(arithm,t2_arithm)]). fof(t5_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), [interesting(0.9),axiom,file(arithm,t5_arithm)]). fof(rqLessOrEqual__r1_xreal_0__r0_r1,theorem,( r1_xreal_0(0,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1)]). fof(rqLessOrEqual__r1_xreal_0__r0_r2,theorem,( r1_xreal_0(0,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2)]). fof(rqLessOrEqual__r1_xreal_0__r0_rn1d2,theorem,( r1_xreal_0(0,k7_xcmplx_0(1,2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rn1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rn1d2)]). fof(rqLessOrEqual__r1_xreal_0__r1_r0,theorem,( ~ r1_xreal_0(1,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0)]). fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r2,theorem,( r1_xreal_0(1,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2)]). fof(rqLessOrEqual__r1_xreal_0__r1_rn1d2,theorem,( ~ r1_xreal_0(1,k7_xcmplx_0(1,2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rn1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rn1d2)]). fof(rqLessOrEqual__r1_xreal_0__r2_r1,theorem,( ~ r1_xreal_0(2,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1)]). fof(rqLessOrEqual__r1_xreal_0__r2_r2,theorem,( r1_xreal_0(2,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2)]). fof(rqLessOrEqual__r1_xreal_0__r2_rn1d2,theorem,( ~ r1_xreal_0(2,k7_xcmplx_0(1,2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rn1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rn1d2)]). fof(rqLessOrEqual__r1_xreal_0__rn1d2_r0,theorem,( ~ r1_xreal_0(k7_xcmplx_0(1,2),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_r0)]). fof(rqLessOrEqual__r1_xreal_0__rn1d2_r1,theorem,( r1_xreal_0(k7_xcmplx_0(1,2),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_r1)]). fof(rqLessOrEqual__r1_xreal_0__rn1d2_r2,theorem,( r1_xreal_0(k7_xcmplx_0(1,2),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_r2)]). fof(rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,theorem,( r1_xreal_0(k7_xcmplx_0(1,2),k7_xcmplx_0(1,2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2)]). fof(rqRealDiv__k7_xcmplx_0__r0_r1_r0,theorem,( k7_xcmplx_0(0,1) = 0 ), file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0)]). fof(rqRealDiv__k7_xcmplx_0__r0_r2_r0,theorem,( k7_xcmplx_0(0,2) = 0 ), file(arithm,rqRealDiv__k7_xcmplx_0__r0_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r0_r2_r0)]). fof(rqRealDiv__k7_xcmplx_0__r1_r1_r1,theorem,( k7_xcmplx_0(1,1) = 1 ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1)]). fof(rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,theorem,( k7_xcmplx_0(1,k7_xcmplx_0(1,2)) = 2 ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2)]). fof(rqRealDiv__k7_xcmplx_0__r2_r1_r2,theorem,( k7_xcmplx_0(2,1) = 2 ), file(arithm,rqRealDiv__k7_xcmplx_0__r2_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r2_r1_r2)]). fof(rqRealDiv__k7_xcmplx_0__r2_r2_r1,theorem,( k7_xcmplx_0(2,2) = 1 ), file(arithm,rqRealDiv__k7_xcmplx_0__r2_r2_r1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r2_r2_r1)]). fof(rqRealMult__k3_xcmplx_0__r0_r0_r0,theorem,( k3_xcmplx_0(0,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r1_r0,theorem,( k3_xcmplx_0(0,1) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r2_r0,theorem,( k3_xcmplx_0(0,2) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r2_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,theorem,( k3_xcmplx_0(0,k7_xcmplx_0(1,2)) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r0_r0,theorem,( k3_xcmplx_0(1,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r1_r1,theorem,( k3_xcmplx_0(1,1) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1)]). fof(rqRealMult__k3_xcmplx_0__r1_r2_r2,theorem,( k3_xcmplx_0(1,2) = 2 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r2_r2)]). fof(rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,theorem,( k3_xcmplx_0(1,k7_xcmplx_0(1,2)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2)]). fof(rqRealMult__k3_xcmplx_0__r2_r0_r0,theorem,( k3_xcmplx_0(2,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r2_r1_r2,theorem,( k3_xcmplx_0(2,1) = 2 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_r1_r2)]). fof(rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,theorem,( k3_xcmplx_0(2,k7_xcmplx_0(1,2)) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1)]). fof(rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,theorem,( k3_xcmplx_0(k7_xcmplx_0(1,2),0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,theorem,( k3_xcmplx_0(k7_xcmplx_0(1,2),2) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(t141_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) & r1_xreal_0(k7_xcmplx_0(A,B),0) ) ) ) ), file(xreal_1,t141_xreal_1), [interesting(0.9),axiom,file(xreal_1,t141_xreal_1)]). fof(rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,theorem,( k7_xcmplx_0(1,2) = k7_xcmplx_0(1,2) ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2)]). fof(rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,theorem,( k3_xcmplx_0(k7_xcmplx_0(1,2),1) = k7_xcmplx_0(1,2) ), file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2)]). fof(rqLessOrEqual__r1_xreal_0__r2_r0,theorem,( ~ r1_xreal_0(2,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0)]). fof(e2_4_1_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_membered,fc30_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc4_arithm,spc7_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k6_real_1,dt_k3_xcmplx_0,dt_k6_real_1,dt_k7_xcmplx_0,dt_c1_4_1_1__bhsp_7,cc2_xreal_0,fc4_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rn1d2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r0,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e1_4_1_1__bhsp_7,t141_xreal_1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r0]), [interesting(0.5),file(bhsp_7,e2_4_1_1__bhsp_7),[file(bhsp_7,e2_4_1_1__bhsp_7)]]). fof(e3_4_1__bhsp_7,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(B,C) & r1_tarski(C,c2_4__bhsp_7) & r1_xreal_0(A,k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7)))) ) ) ) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[dh_c1_4_1__bhsp_7,e2_4_1__bhsp_7]), [interesting(0.65),file(bhsp_7,e3_4_1__bhsp_7),[file(bhsp_7,e3_4_1__bhsp_7)]]). fof(e3_4_1_1__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(A,B) & r1_tarski(B,c2_4__bhsp_7) & r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7)))) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xboole_0,fc23_xreal_0,fc30_xreal_0,fc4_xreal_0,fc5_xreal_0,fc6_membered,fc6_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rn1d2_r0_rn1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rn1d2_r0,spc4_arithm,spc7_arithm,t1_numerals,t1_real,t1_subset,t2_arithm,t3_arithm,t4_arithm,t4_real,t4_subset,t5_arithm,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rn1d2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r0,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc0_boole,spc1_boole,spc2_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e2_4_1_1__bhsp_7,e3_4_1__bhsp_7,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r0]), [interesting(0.5),file(bhsp_7,e3_4_1_1__bhsp_7),[file(bhsp_7,e3_4_1_1__bhsp_7)]]). fof(dt_c2_4_1_1__bhsp_7,plain, ( v1_finset_1(c2_4_1_1__bhsp_7) & m1_subset_1(c2_4_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(consider,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[dh_c2_4_1_1__bhsp_7,e3_4_1_1__bhsp_7]), [interesting(0.5),file(bhsp_7,c2_4_1_1__bhsp_7),[file(bhsp_7,c2_4_1_1__bhsp_7)]]). fof(dh_c1_4_1_1_1__bhsp_7,definition, ( ( ( v1_finset_1(c1_4_1_1_1__bhsp_7) & m1_subset_1(c1_4_1_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(c1_4_1_1_1__bhsp_7) & r1_tarski(c1_4_1_1_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7))) ) ) => ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(c2_4_1_1__bhsp_7,A) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) ), introduced(definition,[new_symbol(c1_4_1_1_1__bhsp_7),file(bhsp_7,c1_4_1_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_1_1_1__bhsp_7)]). fof(e1_4_1_1_1__bhsp_7,assumption, ( ~ v1_xboole_0(c1_4_1_1_1__bhsp_7) & r1_tarski(c1_4_1_1_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7) ), introduced(assumption,[file(bhsp_7,e1_4_1_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,e1_4_1_1_1__bhsp_7)]). fof(fc14_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc14_xreal_0)]). fof(fc24_membered,theorem,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), [interesting(0.9),axiom,file(membered,fc24_membered)]). fof(fc25_membered,theorem,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), [interesting(0.9),axiom,file(membered,fc25_membered)]). fof(fc26_membered,theorem,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), [interesting(0.9),axiom,file(membered,fc26_membered)]). fof(t1_boole,theorem,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), [interesting(0.9),axiom,file(boole,t1_boole)]). fof(commutativity_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(idempotence_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(fc13_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc13_xreal_0)]). fof(fc1_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc1_xreal_0)]). fof(fc22_membered,theorem,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), [interesting(0.9),axiom,file(membered,fc22_membered)]). fof(fc23_membered,theorem,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), [interesting(0.9),axiom,file(membered,fc23_membered)]). fof(fc2_xboole_0,theorem,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ), file(xboole_0,fc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc2_xboole_0)]). fof(fc3_xboole_0,theorem,( ! [A,B] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ), file(xboole_0,fc3_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc3_xboole_0)]). fof(fc9_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), [interesting(0.9),axiom,file(finset_1,fc9_finset_1)]). fof(spc2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,k4_xcmplx_0(1)) = k4_xcmplx_0(A) ) ), file(arithm,spc2_arithm), [interesting(0.9),axiom,file(arithm,spc2_arithm)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(commutativity_k4_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(idempotence_k4_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(redefinition_k4_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(dt_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_c1_4_1_1_1__bhsp_7,assumption, ( v1_finset_1(c1_4_1_1_1__bhsp_7) & m1_subset_1(c1_4_1_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), introduced(assumption,[file(bhsp_7,c1_4_1_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_1_1_1__bhsp_7)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm1,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm2,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r0_rnm1d2,theorem,( ~ r1_xreal_0(0,k7_xcmplx_0(k4_xcmplx_0(1),2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rnm1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rnm1d2)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm1,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm2,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r1_rnm1d2,theorem,( ~ r1_xreal_0(1,k7_xcmplx_0(k4_xcmplx_0(1),2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rnm1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rnm1d2)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm1,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm2,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r2_rnm1d2,theorem,( ~ r1_xreal_0(2,k7_xcmplx_0(k4_xcmplx_0(1),2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rnm1d2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r2,theorem,( r1_xreal_0(k4_xcmplx_0(1),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm2,theorem,( ~ r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rn1d2,theorem,( r1_xreal_0(k4_xcmplx_0(1),k7_xcmplx_0(1,2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rn1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rn1d2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rnm1d2,theorem,( r1_xreal_0(k4_xcmplx_0(1),k7_xcmplx_0(k4_xcmplx_0(1),2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r0,theorem,( r1_xreal_0(k4_xcmplx_0(2),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r1,theorem,( r1_xreal_0(k4_xcmplx_0(2),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r2,theorem,( r1_xreal_0(k4_xcmplx_0(2),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm2,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rn1d2_rm1,theorem,( ~ r1_xreal_0(k7_xcmplx_0(1,2),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2,theorem,( ~ r1_xreal_0(k7_xcmplx_0(1,2),k7_xcmplx_0(k4_xcmplx_0(1),2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2)]). fof(rqLessOrEqual__r1_xreal_0__rnm1d2_r0,theorem,( r1_xreal_0(k7_xcmplx_0(k4_xcmplx_0(1),2),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_r0)]). fof(rqLessOrEqual__r1_xreal_0__rnm1d2_r1,theorem,( r1_xreal_0(k7_xcmplx_0(k4_xcmplx_0(1),2),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_r1)]). fof(rqLessOrEqual__r1_xreal_0__rnm1d2_r2,theorem,( r1_xreal_0(k7_xcmplx_0(k4_xcmplx_0(1),2),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_r2)]). fof(rqLessOrEqual__r1_xreal_0__rnm1d2_rm1,theorem,( ~ r1_xreal_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2,theorem,( r1_xreal_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k7_xcmplx_0(1,2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2)]). fof(rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2,theorem,( r1_xreal_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k7_xcmplx_0(k4_xcmplx_0(1),2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2)]). fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,( k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r0_r2_rm2,theorem,( k6_xcmplx_0(0,2) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm2_r2,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2)]). fof(rqRealDiff__k6_xcmplx_0__r0_rn1d2_rnm1d2,theorem,( k6_xcmplx_0(0,k7_xcmplx_0(1,2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rn1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rn1d2_rnm1d2)]). fof(rqRealDiff__k6_xcmplx_0__r0_rnm1d2_rn1d2,theorem,( k6_xcmplx_0(0,k7_xcmplx_0(k4_xcmplx_0(1),2)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rnm1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rnm1d2_rn1d2)]). fof(rqRealDiff__k6_xcmplx_0__r1_r2_rm1,theorem,( k6_xcmplx_0(1,2) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k7_xcmplx_0(k4_xcmplx_0(1),2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2)]). fof(rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0)]). fof(rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,theorem,( k6_xcmplx_0(k7_xcmplx_0(1,2),1) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2)]). fof(rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,theorem,( k6_xcmplx_0(k7_xcmplx_0(1,2),k7_xcmplx_0(k4_xcmplx_0(1),2)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1)]). fof(rqRealDiff__k6_xcmplx_0__rnm1d2_r0_rnm1d2,theorem,( k6_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),0) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_r0_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_r0_rnm1d2)]). fof(rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,theorem,( k6_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k4_xcmplx_0(1)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2)]). fof(rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,theorem,( k6_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k7_xcmplx_0(1,2)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rnm1d2_rnm1d2_r0,theorem,( k6_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k7_xcmplx_0(k4_xcmplx_0(1),2)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_rnm1d2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rnm1d2_rnm1d2_r0)]). fof(rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,theorem,( k7_xcmplx_0(1,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1)]). fof(rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,theorem,( k7_xcmplx_0(1,k4_xcmplx_0(2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2)]). fof(rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,theorem,( k7_xcmplx_0(1,k7_xcmplx_0(k4_xcmplx_0(1),2)) = k4_xcmplx_0(2) ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2)]). fof(rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,theorem,( k7_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1)]). fof(rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,theorem,( k7_xcmplx_0(k4_xcmplx_0(1),2) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2)]). fof(rqRealMult__k3_xcmplx_0__r0_rm2_r0,theorem,( k3_xcmplx_0(0,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_rm2_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_rm2_rm2,theorem,( k3_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealMult__k3_xcmplx_0__r1_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_rm2_rm2)]). fof(rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2,theorem,( k3_xcmplx_0(1,k7_xcmplx_0(k4_xcmplx_0(1),2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2)]). fof(rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1,theorem,( k3_xcmplx_0(2,k7_xcmplx_0(k4_xcmplx_0(1),2)) = k4_xcmplx_0(1) ), file(arithm,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1)]). fof(rqRealMult__k3_xcmplx_0__rm2_r0_r0,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__rm2_r1_rm2,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(2) ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_r1_rm2)]). fof(rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),k7_xcmplx_0(1,2)) = k4_xcmplx_0(1) ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1)]). fof(rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),k7_xcmplx_0(k4_xcmplx_0(1),2)) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1)]). fof(rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1,theorem,( k3_xcmplx_0(k7_xcmplx_0(1,2),k4_xcmplx_0(2)) = k4_xcmplx_0(1) ), file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1)]). fof(rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2,theorem,( k3_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),1) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2)]). fof(rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1,theorem,( k3_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),2) = k4_xcmplx_0(1) ), file(arithm,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1)]). fof(rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1,theorem,( k3_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k4_xcmplx_0(2)) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(rqRealNeg__k4_xcmplx_0__rm2_r2,theorem,( k4_xcmplx_0(k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2)]). fof(rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,theorem,( k4_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2)]). fof(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). fof(spc1_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,k4_xcmplx_0(B)) = k6_xcmplx_0(A,B) ) ), file(arithm,spc1_arithm), [interesting(0.9),axiom,file(arithm,spc1_arithm)]). fof(spc6_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k2_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(A,k2_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(t1_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), [interesting(0.9),axiom,file(arithm,t1_arithm)]). fof(commutativity_k2_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(commutativity_k3_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(redefinition_k3_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r0_r0,theorem,( k2_xcmplx_0(0,0) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r2_r2,theorem,( k2_xcmplx_0(0,2) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__r1_r1_r2,theorem,( k2_xcmplx_0(1,1) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm1_r0,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r2_r0_r2,theorem,( k2_xcmplx_0(2,0) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm1_r1,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm2_r0,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r2_r1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),2) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r2_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),2) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0)]). fof(rqRealNeg__k4_xcmplx_0__r2_rm2,theorem,( k4_xcmplx_0(2) = k4_xcmplx_0(2) ), file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2)]). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), [interesting(0.9),axiom,file(binop_1,k1_binop_1)]). fof(redefinition_k2_binop_1,definition,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ), file(binop_1,k2_binop_1), [interesting(0.9),axiom,file(binop_1,k2_binop_1)]). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), [interesting(0.9),axiom,file(relat_1,k1_relat_1)]). fof(dt_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ), file(binop_1,k2_binop_1), [interesting(0.9),axiom,file(binop_1,k2_binop_1)]). fof(redefinition_k4_relset_1,definition,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [interesting(0.9),axiom,file(funct_2,d1_funct_2)]). fof(e6_4_1_1_1__bhsp_7,plain,( k1_relat_1(c3_4__bhsp_7) = u1_struct_0(c1_4__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_numbers,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc6_membered,cc7_xreal_0,fc14_finset_1,fc2_membered,rc1_finset_1,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,redefinition_k6_supinf_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc1_struct_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k4_relset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,fc1_xboole_0,fc6_membered,t6_boole,d1_funct_2]), [interesting(0.35),file(bhsp_7,e6_4_1_1_1__bhsp_7),[file(bhsp_7,e6_4_1_1_1__bhsp_7)]]). fof(t14_bhsp_5,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m2_relset_1(C,k2_zfmisc_1(A,A),A) ) => ( ( v1_binop_1(C,A) & v2_binop_1(C,A) & v1_setwiseo(C,A) ) => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(B)) ) => ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(B)) ) => ( r1_xboole_0(D,E) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,B,A) & m2_relset_1(F,B,A) ) => ( ( r1_tarski(D,k1_relat_1(F)) & r1_tarski(E,k1_relat_1(F)) ) => ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(B)) ) => ( G = k2_xboole_0(D,E) => k5_bhsp_5(A,B,C,G,F) = k2_binop_1(A,A,A,C,k5_bhsp_5(A,B,C,D,F),k5_bhsp_5(A,B,C,E,F)) ) ) ) ) ) ) ) ) ) ) ) ), file(bhsp_5,t14_bhsp_5), [interesting(0.9),axiom,file(bhsp_5,t14_bhsp_5)]). fof(e1_4_1_1_1_1__bhsp_7,plain,( k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),c3_4__bhsp_7) = k2_binop_1(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c2_4_1_1__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c2_4__bhsp_7,e1_4_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1_1_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc3_funct_1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k1_binop_1,dt_k1_xboole_0,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_binop_1,redefinition_k4_subset_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_binop_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k4_subset_1,dt_k5_bhsp_5,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc14_finset_1,fc2_membered,fc2_xboole_0,fc3_binop_2,fc3_xboole_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,e6_4_1_1_1__bhsp_7,e1_4_1_1_1__bhsp_7,t14_bhsp_5]), [interesting(0.2),file(bhsp_7,e1_4_1_1_1_1__bhsp_7),[file(bhsp_7,e1_4_1_1_1_1__bhsp_7)]]). fof(commutativity_k9_binop_2,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k9_binop_2(B,A) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(redefinition_k9_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(dt_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k9_binop_2(A,B),k1_numbers) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(d9_binop_2,definition,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ) => ( A = k33_binop_2 <=> ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k9_binop_2(B,C) ) ) ) ) ), file(binop_2,d9_binop_2), [interesting(0.9),axiom,file(binop_2,d9_binop_2)]). fof(e2_4_1_1_1_1__bhsp_7,plain,( k2_binop_1(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c2_4_1_1__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7)) = k3_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c2_4_1_1__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_1__bhsp_7,dt_c3_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc3_xreal_0,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,spc6_arithm,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k1_binop_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,fc14_finset_1,fc1_struct_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,commutativity_k9_binop_2,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_binop_1,redefinition_k3_real_1,redefinition_k9_binop_2,redefinition_m2_relset_1,dt_k1_numbers,dt_k2_binop_1,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k9_binop_2,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,d9_binop_2]), [interesting(0.2),file(bhsp_7,e2_4_1_1_1_1__bhsp_7),[file(bhsp_7,e2_4_1_1_1_1__bhsp_7)]]). fof(e7_4_1_1_1__bhsp_7,plain,( k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),c3_4__bhsp_7) = k3_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c2_4_1_1__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7)) ), inference(iterative_eq,[status(thm),assumptions([e1_4_1_1_1__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_1__bhsp_7,dt_c3_4__bhsp_7])],[e1_4_1_1_1_1__bhsp_7,e2_4_1_1_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e7_4_1_1_1__bhsp_7),[file(bhsp_7,e7_4_1_1_1__bhsp_7)]]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_rm1_r2,theorem,( k6_xcmplx_0(1,k4_xcmplx_0(1)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r1_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),1) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0)]). fof(e8_4_1_1_1__bhsp_7,plain,( k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7) = k5_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c2_4_1_1__bhsp_7,c3_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([e1_4_1_1_1__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_1__bhsp_7,dt_c3_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc20_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_xboole_0,fc1_xreal_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc5_membered,fc5_xreal_0,fc6_membered,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc14_finset_1,fc1_struct_0,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k4_subset_1,idempotence_k4_subset_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k4_subset_1,redefinition_k5_real_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k33_binop_2,dt_k3_real_1,dt_k4_subset_1,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_xcmplx_0,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e7_4_1_1_1__bhsp_7,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0]), [interesting(0.35),file(bhsp_7,e8_4_1_1_1__bhsp_7),[file(bhsp_7,e8_4_1_1_1__bhsp_7)]]). fof(rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,theorem,( k4_xcmplx_0(k7_xcmplx_0(1,2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2)]). fof(e4_4_1_1__bhsp_7,plain, ( ~ v1_xboole_0(c2_4_1_1__bhsp_7) & r1_tarski(c2_4_1_1__bhsp_7,c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( r1_tarski(c2_4_1_1__bhsp_7,A) & r1_tarski(A,c2_4__bhsp_7) & r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7)))) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[dh_c2_4_1_1__bhsp_7,e3_4_1_1__bhsp_7]), [interesting(0.5),file(bhsp_7,e4_4_1_1__bhsp_7),[file(bhsp_7,e4_4_1_1__bhsp_7)]]). fof(t7_xboole_1,theorem,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), [interesting(0.9),axiom,file(xboole_1,t7_xboole_1)]). fof(t8_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), [interesting(0.9),axiom,file(xboole_1,t8_xboole_1)]). fof(e2_4_1_1_1__bhsp_7,plain, ( k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7) = k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7) & r1_tarski(c2_4_1_1__bhsp_7,k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7)) & r1_tarski(k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),c2_4__bhsp_7) & v1_finset_1(k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7)) & m1_subset_1(k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7,e1_4_1_1_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc23_xreal_0,fc24_membered,fc25_membered,fc26_membered,fc30_xreal_0,fc4_xreal_0,fc5_xreal_0,fc6_membered,fc6_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,spc4_arithm,spc7_arithm,t1_boole,t1_real,t1_subset,t3_arithm,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xcmplx_0,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k4_subset_1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k4_subset_1,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc2_xboole_0,fc3_binop_2,fc3_xboole_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc1_boole,spc2_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e4_4_1_1__bhsp_7,e1_4_1_1_1__bhsp_7,t7_xboole_1,t8_xboole_1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2]), [interesting(0.35),file(bhsp_7,e2_4_1_1_1__bhsp_7),[file(bhsp_7,e2_4_1_1_1__bhsp_7)]]). fof(e3_4_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),c3_4__bhsp_7)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,e1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xboole_0,fc1_xreal_0,fc22_membered,fc23_membered,fc23_xreal_0,fc24_membered,fc25_membered,fc26_membered,fc2_xboole_0,fc30_xreal_0,fc3_xboole_0,fc4_xreal_0,fc5_xreal_0,fc6_membered,fc6_xreal_0,fc9_finset_1,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_boole,t1_real,t1_subset,t3_arithm,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_subset_1,idempotence_k4_subset_1,involutiveness_k4_xcmplx_0,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k4_subset_1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k4_subset_1,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r1_rnm1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rnm1d2,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm1_rn1d2,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rm1,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_r1,rqLessOrEqual__r1_xreal_0__rnm1d2_r2,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,spc1_boole,spc2_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e2_4_1_1_1__bhsp_7,e4_4_1_1__bhsp_7,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(bhsp_7,e3_4_1_1_1__bhsp_7),[file(bhsp_7,e3_4_1_1_1__bhsp_7)]]). fof(t13_uniform1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => k18_complex1(k5_real_1(A,B)) = k18_complex1(k5_real_1(B,A)) ) ) ), file(uniform1,t13_uniform1), [interesting(0.9),axiom,file(uniform1,t13_uniform1)]). fof(e4_4_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,k4_subset_1(u1_struct_0(c1_4__bhsp_7),c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7),c3_4__bhsp_7),c1_4_1__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,e1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_membered,fc24_xreal_0,fc25_membered,fc26_membered,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_boole,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc22_membered,fc23_membered,fc23_xreal_0,fc2_xboole_0,fc30_xreal_0,fc3_xboole_0,fc4_xreal_0,fc5_xreal_0,fc6_xreal_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_subset_1,idempotence_k4_subset_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k4_subset_1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k4_subset_1,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r1_rnm1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rnm1d2,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm1_rn1d2,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rm1,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_r1,rqLessOrEqual__r1_xreal_0__rnm1d2_r2,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e3_4_1_1_1__bhsp_7,t13_uniform1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(bhsp_7,e4_4_1_1_1__bhsp_7),[file(bhsp_7,e4_4_1_1_1__bhsp_7)]]). fof(e5_4_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(c1_4_1_1__bhsp_7,2),k18_complex1(k5_real_1(c1_4_1__bhsp_7,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c2_4_1_1__bhsp_7,c3_4__bhsp_7)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xboole_0,fc1_xreal_0,fc23_xreal_0,fc30_xreal_0,fc4_xreal_0,fc5_xreal_0,fc6_membered,fc6_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_real,t1_subset,t3_arithm,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r1_rnm1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rnm1d2,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm1_rn1d2,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rm1,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_r1,rqLessOrEqual__r1_xreal_0__rnm1d2_r2,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,spc1_boole,spc2_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e4_4_1_1__bhsp_7,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(bhsp_7,e5_4_1_1_1__bhsp_7),[file(bhsp_7,e5_4_1_1_1__bhsp_7)]]). fof(l1_bhsp_7,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( ~ r1_xreal_0(k6_real_1(D,2),k18_complex1(k5_real_1(A,B))) & ~ r1_xreal_0(k6_real_1(D,2),k18_complex1(k5_real_1(B,C))) & r1_xreal_0(D,k18_complex1(k5_real_1(A,C))) ) ) ) ) ) ), file(bhsp_7,l1_bhsp_7), [interesting(0.9),axiom,file(bhsp_7,l1_bhsp_7)]). fof(e9_4_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,e1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_membered,fc24_xreal_0,fc25_membered,fc26_membered,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_boole,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc22_membered,fc23_membered,fc23_xreal_0,fc2_xboole_0,fc30_xreal_0,fc3_xboole_0,fc4_xreal_0,fc5_xreal_0,fc6_xreal_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_subset_1,idempotence_k4_subset_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k4_subset_1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k4_subset_1,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c1_4_1_1_1__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r0_rn1d2,rqLessOrEqual__r1_xreal_0__r0_rnm1d2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r1_rnm1d2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rnm1d2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm1_rn1d2,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqLessOrEqual__r1_xreal_0__rn1d2_r0,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rm1,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_r0,rqLessOrEqual__r1_xreal_0__rnm1d2_r1,rqLessOrEqual__r1_xreal_0__rnm1d2_r2,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r0_rn1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__r0_rnm1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealDiff__k6_xcmplx_0__rn1d2_r0_rn1d2,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rn1d2_r0,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,rqRealDiff__k6_xcmplx_0__rnm1d2_r0_rnm1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,rqRealDiff__k6_xcmplx_0__rnm1d2_rnm1d2_r0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e8_4_1_1_1__bhsp_7,e4_4_1_1_1__bhsp_7,e5_4_1_1_1__bhsp_7,l1_bhsp_7,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.35),file(bhsp_7,e9_4_1_1_1__bhsp_7),[file(bhsp_7,e9_4_1_1_1__bhsp_7)]]). fof(i3_4_1_1_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_1_1_1__bhsp_7)]), [interesting(0.35),trivial,file(bhsp_7,i3_4_1_1_1__bhsp_7)]). fof(i2_4_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,e1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[e9_4_1_1_1__bhsp_7,i3_4_1_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,i2_4_1_1_1__bhsp_7),[file(bhsp_7,i2_4_1_1_1__bhsp_7)]]). fof(i1_4_1_1_1__bhsp_7,plain,( ~ ( ~ v1_xboole_0(c1_4_1_1_1__bhsp_7) & r1_tarski(c1_4_1_1_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_1_1_1__bhsp_7,dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7]),discharge_asm(discharge,[e1_4_1_1_1__bhsp_7])],[e1_4_1_1_1__bhsp_7,i2_4_1_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,i1_4_1_1_1__bhsp_7),[file(bhsp_7,i1_4_1_1_1__bhsp_7)]]). fof(i1_4_1_1_1_tmp__bhsp_7,plain, ( ( v1_finset_1(c1_4_1_1_1__bhsp_7) & m1_subset_1(c1_4_1_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(c1_4_1_1_1__bhsp_7) & r1_tarski(c1_4_1_1_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c2_4_1_1__bhsp_7,c1_4_1_1_1__bhsp_7) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_1_1_1__bhsp_7,c3_4__bhsp_7))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7]),discharge_asm(discharge,[dt_c1_4_1_1_1__bhsp_7])],[dt_c1_4_1_1_1__bhsp_7,i1_4_1_1_1__bhsp_7]), [interesting(0.5),e5_4_1_1__bhsp_7]). fof(e5_4_1_1__bhsp_7,plain,( ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(c2_4_1_1__bhsp_7,A) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[i1_4_1_1_1_tmp__bhsp_7,dh_c1_4_1_1_1__bhsp_7]), [interesting(0.5),file(bhsp_7,e5_4_1_1__bhsp_7),[file(bhsp_7,e5_4_1_1__bhsp_7)]]). fof(e6_4_1_1__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xboole_0,fc23_xreal_0,fc30_xreal_0,fc4_xreal_0,fc5_xreal_0,fc6_membered,fc6_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,spc4_arithm,spc7_arithm,t1_real,t1_subset,t3_arithm,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k5_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k5_bhsp_5,dt_k5_real_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_1__bhsp_7,dt_c1_4_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_1_1__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc1_boole,spc2_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e5_4_1_1__bhsp_7,e4_4_1_1__bhsp_7,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2]), [interesting(0.5),file(bhsp_7,e6_4_1_1__bhsp_7),[file(bhsp_7,e6_4_1_1__bhsp_7)]]). fof(i3_4_1_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_1_1__bhsp_7)]), [interesting(0.5),trivial,file(bhsp_7,i3_4_1_1__bhsp_7)]). fof(i2_4_1_1__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,e1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[e6_4_1_1__bhsp_7,i3_4_1_1__bhsp_7]), [interesting(0.5),file(bhsp_7,i2_4_1_1__bhsp_7),[file(bhsp_7,i2_4_1_1__bhsp_7)]]). fof(i1_4_1_1__bhsp_7,plain,( ~ ( ~ r1_xreal_0(c1_4_1_1__bhsp_7,0) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7]),discharge_asm(discharge,[e1_4_1_1__bhsp_7])],[e1_4_1_1__bhsp_7,i2_4_1_1__bhsp_7]), [interesting(0.5),file(bhsp_7,i1_4_1_1__bhsp_7),[file(bhsp_7,i1_4_1_1__bhsp_7)]]). fof(i1_4_1_1_tmp__bhsp_7,plain, ( m1_subset_1(c1_4_1_1__bhsp_7,k1_numbers) => ~ ( ~ r1_xreal_0(c1_4_1_1__bhsp_7,0) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c1_4_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7]),discharge_asm(discharge,[dt_c1_4_1_1__bhsp_7])],[dt_c1_4_1_1__bhsp_7,i1_4_1_1__bhsp_7]), [interesting(0.65),e4_4_1__bhsp_7]). fof(e4_4_1__bhsp_7,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(B,C) & r1_xreal_0(A,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[i1_4_1_1_tmp__bhsp_7,dh_c1_4_1_1__bhsp_7]), [interesting(0.65),file(bhsp_7,e4_4_1__bhsp_7),[file(bhsp_7,e4_4_1__bhsp_7)]]). fof(e5_4_1__bhsp_7,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(B,C) & r1_xreal_0(A,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc5_membered,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc6_membered,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,t1_numerals,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t8_boole,projectivity_k18_complex1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc0_boole,spc0_numerals,e4_4_1__bhsp_7]), [interesting(0.65),file(bhsp_7,e5_4_1__bhsp_7),[file(bhsp_7,e5_4_1__bhsp_7)]]). fof(i2_4_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i2_4_1__bhsp_7)]), [interesting(0.65),trivial,file(bhsp_7,i2_4_1__bhsp_7)]). fof(i1_4_1__bhsp_7,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(B,C) & r1_xreal_0(A,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_1__bhsp_7])],[e5_4_1__bhsp_7,i2_4_1__bhsp_7]), [interesting(0.65),file(bhsp_7,i1_4_1__bhsp_7),[file(bhsp_7,i1_4_1__bhsp_7)]]). fof(e1_4__bhsp_7,plain, ( r1_bhsp_6(c1_4__bhsp_7,c2_4__bhsp_7,c3_4__bhsp_7) => ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(B,C) & r1_xreal_0(A,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7]),discharge_asm(discharge,[e1_4_1__bhsp_7])],[e1_4_1__bhsp_7,i1_4_1__bhsp_7]), [interesting(0.8),file(bhsp_7,e1_4__bhsp_7),[file(bhsp_7,e1_4__bhsp_7)]]). fof(e2_4__bhsp_7,assumption,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(B,C) & r1_xreal_0(A,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), introduced(assumption,[file(bhsp_7,e2_4__bhsp_7)]), [interesting(0.8),axiom,file(bhsp_7,e2_4__bhsp_7)]). fof(dh_c4_4_2__bhsp_7,definition, ( ? [A] : ( v1_xreal_0(A) & ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & r1_xreal_0(C,D) & r1_xreal_0(B,k18_complex1(k6_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,D),A))) ) ) ) ) ) => ( v1_xreal_0(c4_4_2__bhsp_7) & ! [E] : ( v1_xreal_0(E) => ~ ( ~ r1_xreal_0(E,0) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ? [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) & r1_xreal_0(F,G) & r1_xreal_0(E,k18_complex1(k6_xcmplx_0(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,G),c4_4_2__bhsp_7))) ) ) ) ) ) ), introduced(definition,[new_symbol(c4_4_2__bhsp_7),file(bhsp_7,c4_4_2__bhsp_7)]), [interesting(0.65),axiom,file(bhsp_7,c4_4_2__bhsp_7)]). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(redefinition_k2_seq_1,definition,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(redefinition_k8_funct_2,definition,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), [interesting(0.9),axiom,file(funct_2,k8_funct_2)]). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), [interesting(0.9),axiom,file(funct_2,k8_funct_2)]). fof(dh_c2_4_2__bhsp_7,definition, ( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ~ ( r2_hidden(B,k5_numbers) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & k1_funct_1(c1_4_2__bhsp_7,B) = C & k1_funct_1(A,B) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7) ) ) ) ) => ( v1_funct_1(c2_4_2__bhsp_7) & v1_funct_2(c2_4_2__bhsp_7,k5_numbers,k1_numbers) & m2_relset_1(c2_4_2__bhsp_7,k5_numbers,k1_numbers) & ! [D] : ~ ( r2_hidden(D,k5_numbers) & ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(E) & k1_funct_1(c1_4_2__bhsp_7,D) = E & k1_funct_1(c2_4_2__bhsp_7,D) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,E,c3_4__bhsp_7) ) ) ) ) ), introduced(definition,[new_symbol(c2_4_2__bhsp_7),file(bhsp_7,c2_4_2__bhsp_7)]), [interesting(0.65),axiom,file(bhsp_7,c2_4_2__bhsp_7)]). fof(dh_c1_4_2__bhsp_7,definition, ( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),C) & r1_xreal_0(k6_real_1(1,k1_nat_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,C)) ) ) ) ) ) => ( v1_funct_1(c1_4_2__bhsp_7) & v1_funct_2(c1_4_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(c1_4_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,D)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,D),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,D)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,D),c2_4__bhsp_7) & ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(E) & r1_tarski(E,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,D),E) & r1_xreal_0(k6_real_1(1,k1_nat_1(D,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,E,c3_4__bhsp_7))) ) ) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_xreal_0(D,E) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,D),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,E)) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2__bhsp_7),file(bhsp_7,c1_4_2__bhsp_7)]), [interesting(0.65),axiom,file(bhsp_7,c1_4_2__bhsp_7)]). fof(commutativity_k1_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(redefinition_k1_nat_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dh_c2_4_2_2__bhsp_7,definition, ( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = k5_numbers & k1_funct_1(A,0) = k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_funct_1(A,k1_nat_1(B,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(B,1)),k1_funct_1(A,B)) ) ) => ( v1_relat_1(c2_4_2_2__bhsp_7) & v1_funct_1(c2_4_2_2__bhsp_7) & k1_relat_1(c2_4_2_2__bhsp_7) = k5_numbers & k1_funct_1(c2_4_2_2__bhsp_7,0) = k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(C,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(C,1)),k1_funct_1(c2_4_2_2__bhsp_7,C)) ) ) ), introduced(definition,[new_symbol(c2_4_2_2__bhsp_7),file(bhsp_7,c2_4_2_2__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c2_4_2_2__bhsp_7)]). fof(dh_c1_4_2_2__bhsp_7,definition, ( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( r2_hidden(B,k5_numbers) => ( v1_finset_1(k1_funct_1(A,B)) & m1_subset_1(k1_funct_1(A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(A,B)) & r1_tarski(k1_funct_1(A,B),c2_4__bhsp_7) & ! [C] : ( m1_subset_1(C,k1_numbers) => ( C = B => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(A,B),D) & r1_xreal_0(k6_real_1(1,k3_real_1(C,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7))) ) ) ) ) ) ) ) => ( v1_funct_1(c1_4_2_2__bhsp_7) & v1_funct_2(c1_4_2_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(c1_4_2_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [E] : ( r2_hidden(E,k5_numbers) => ( v1_finset_1(k1_funct_1(c1_4_2_2__bhsp_7,E)) & m1_subset_1(k1_funct_1(c1_4_2_2__bhsp_7,E),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c1_4_2_2__bhsp_7,E)) & r1_tarski(k1_funct_1(c1_4_2_2__bhsp_7,E),c2_4__bhsp_7) & ! [F] : ( m1_subset_1(F,k1_numbers) => ( F = E => ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(G) & r1_tarski(G,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c1_4_2_2__bhsp_7,E),G) & r1_xreal_0(k6_real_1(1,k3_real_1(F,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,G,c3_4__bhsp_7))) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_2__bhsp_7),file(bhsp_7,c1_4_2_2__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_2__bhsp_7)]). fof(s1_funct_2__e3_4_2_1__bhsp_7,theorem,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k6_supinf_1) & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) ) => ( ! [D] : ~ ( r2_hidden(D,k5_numbers) & ! [E] : ~ ( r2_hidden(E,k1_zfmisc_1(B)) & v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(E) & r1_tarski(E,B) & ! [F] : ( m1_subset_1(F,k1_numbers) => ( F = D => ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(G) & r1_tarski(G,B) & r1_xboole_0(E,G) & r1_xreal_0(k6_real_1(1,k3_real_1(F,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C))) ) ) ) ) ) ) => ? [D] : ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_zfmisc_1(B)) & m2_relset_1(D,k5_numbers,k1_zfmisc_1(B)) & ! [E] : ( r2_hidden(E,k5_numbers) => ( v1_finset_1(k1_funct_1(D,E)) & m1_subset_1(k1_funct_1(D,E),k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(k1_funct_1(D,E)) & r1_tarski(k1_funct_1(D,E),B) & ! [H] : ( m1_subset_1(H,k1_numbers) => ( H = E => ! [I] : ( ( v1_finset_1(I) & m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(I) & r1_tarski(I,B) & r1_xboole_0(k1_funct_1(D,E),I) & r1_xreal_0(k6_real_1(1,k3_real_1(H,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,I,C))) ) ) ) ) ) ) ) ) ) ), file(bhsp_7,s1_funct_2__e3_4_2_1__bhsp_7), [interesting(0.9),axiom,file(bhsp_7,s1_funct_2__e3_4_2_1__bhsp_7)]). fof(dh_c1_4_2_1_1__bhsp_7,definition, ( ~ ( r2_hidden(c1_4_2_1_1__bhsp_7,k5_numbers) & ! [A] : ~ ( r2_hidden(A,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( m1_subset_1(B,k1_numbers) => ( B = c1_4_2_1_1__bhsp_7 => ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(A,C) & r1_xreal_0(k6_real_1(1,k3_real_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) => ! [D] : ~ ( r2_hidden(D,k5_numbers) & ! [E] : ~ ( r2_hidden(E,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(E) & r1_tarski(E,c2_4__bhsp_7) & ! [F] : ( m1_subset_1(F,k1_numbers) => ( F = D => ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(G) & r1_tarski(G,c2_4__bhsp_7) & r1_xboole_0(E,G) & r1_xreal_0(k6_real_1(1,k3_real_1(F,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,G,c3_4__bhsp_7))) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_1_1__bhsp_7),file(bhsp_7,c1_4_2_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_1_1__bhsp_7)]). fof(e1_4_2_1_1__bhsp_7,assumption,( r2_hidden(c1_4_2_1_1__bhsp_7,k5_numbers) ), introduced(assumption,[file(bhsp_7,e1_4_2_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,e1_4_2_1_1__bhsp_7)]). fof(dt_c1_4_2_1_1__bhsp_7,assumption,( $true ), introduced(assumption,[file(bhsp_7,c1_4_2_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_1_1__bhsp_7)]). fof(de_c2_4_2_1_1__bhsp_7,definition,( c2_4_2_1_1__bhsp_7 = c1_4_2_1_1__bhsp_7 ), introduced(definition,[new_symbol(c2_4_2_1_1__bhsp_7),file(bhsp_7,c2_4_2_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c2_4_2_1_1__bhsp_7)]). fof(e2_4_2_1_1__bhsp_7,plain,( m2_subset_1(c1_4_2_1_1__bhsp_7,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,fc2_membered,t1_subset,t7_boole,e1_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e2_4_2_1_1__bhsp_7),[file(bhsp_7,e2_4_2_1_1__bhsp_7)]]). fof(dt_c2_4_2_1_1__bhsp_7,plain,( m2_subset_1(c2_4_2_1_1__bhsp_7,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,fc2_membered,de_c2_4_2_1_1__bhsp_7,e2_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,c2_4_2_1_1__bhsp_7),[file(bhsp_7,c2_4_2_1_1__bhsp_7)]]). fof(de_c3_4_2_1_1__bhsp_7,definition,( c3_4_2_1_1__bhsp_7 = k6_real_1(1,k1_nat_1(c2_4_2_1_1__bhsp_7,1)) ), introduced(definition,[new_symbol(c3_4_2_1_1__bhsp_7),file(bhsp_7,c3_4_2_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c3_4_2_1_1__bhsp_7)]). fof(e3_4_2_1_1__bhsp_7,plain,( m1_subset_1(k6_real_1(1,k1_nat_1(c2_4_2_1_1__bhsp_7,1)),k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,t6_arithm,commutativity_k2_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_funct_1,cc4_membered,rc1_xboole_0,rc2_xboole_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,existence_m1_subset_1,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k1_nat_1,dt_k1_numbers,dt_k6_real_1,dt_m1_subset_1,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,fc2_membered,spc1_boole,spc1_numerals]), [interesting(0.35),file(bhsp_7,e3_4_2_1_1__bhsp_7),[file(bhsp_7,e3_4_2_1_1__bhsp_7)]]). fof(dt_c3_4_2_1_1__bhsp_7,plain,( m1_subset_1(c3_4_2_1_1__bhsp_7,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,t6_arithm,commutativity_k2_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_funct_1,cc4_membered,rc1_xboole_0,rc2_xboole_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,existence_m1_subset_1,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k1_nat_1,dt_k1_numbers,dt_k6_real_1,dt_m1_subset_1,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,fc2_membered,spc1_boole,spc1_numerals,de_c3_4_2_1_1__bhsp_7,e3_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,c3_4_2_1_1__bhsp_7),[file(bhsp_7,c3_4_2_1_1__bhsp_7)]]). fof(dh_c4_4_2_1_1__bhsp_7,definition, ( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c3_4_2_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) => ( v1_finset_1(c4_4_2_1_1__bhsp_7) & m1_subset_1(c4_4_2_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(c4_4_2_1_1__bhsp_7) & r1_tarski(c4_4_2_1_1__bhsp_7,c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(c4_4_2_1_1__bhsp_7,C) & r1_xreal_0(c3_4_2_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ), introduced(definition,[new_symbol(c4_4_2_1_1__bhsp_7),file(bhsp_7,c4_4_2_1_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c4_4_2_1_1__bhsp_7)]). fof(t18_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_1), [interesting(0.9),axiom,file(nat_1,t18_nat_1)]). fof(e4_4_2_1_1__bhsp_7,plain,( r1_xreal_0(0,c2_4_2_1_1__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,rc1_xboole_0,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_boole,spc0_numerals,t18_nat_1]), [interesting(0.35),file(bhsp_7,e4_4_2_1_1__bhsp_7),[file(bhsp_7,e4_4_2_1_1__bhsp_7)]]). fof(t38_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), [interesting(0.9),axiom,file(nat_1,t38_nat_1)]). fof(e5_4_2_1_1__bhsp_7,plain,( ~ r1_xreal_0(k1_nat_1(c2_4_2_1_1__bhsp_7,1),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r0,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e4_4_2_1_1__bhsp_7,t38_nat_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(bhsp_7,e5_4_2_1_1__bhsp_7),[file(bhsp_7,e5_4_2_1_1__bhsp_7)]]). fof(t73_real_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(A,0) => ( ~ ( ~ r1_xreal_0(C,B) & r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) ) & ~ ( ~ r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) & r1_xreal_0(C,B) ) ) ) ) ) ) ), file(real_1,t73_real_1), [interesting(0.9),axiom,file(real_1,t73_real_1)]). fof(e6_4_2_1_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(1,k1_nat_1(c2_4_2_1_1__bhsp_7,1)),k6_real_1(0,k1_nat_1(c2_4_2_1_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc30_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k1_nat_1,dt_k6_real_1,dt_k7_xcmplx_0,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,cc2_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e5_4_2_1_1__bhsp_7,t73_real_1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1]), [interesting(0.35),file(bhsp_7,e6_4_2_1_1__bhsp_7),[file(bhsp_7,e6_4_2_1_1__bhsp_7)]]). fof(e7_4_2_1_1__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(A,B) & r1_xreal_0(c3_4_2_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t1_subset,t4_real,t4_subset,t5_arithm,t5_subset,t6_arithm,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k6_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_1_1__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,de_c3_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,spc0_boole,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e6_4_2_1_1__bhsp_7,e2_4__bhsp_7,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.35),file(bhsp_7,e7_4_2_1_1__bhsp_7),[file(bhsp_7,e7_4_2_1_1__bhsp_7)]]). fof(dt_c4_4_2_1_1__bhsp_7,plain, ( v1_finset_1(c4_4_2_1_1__bhsp_7) & m1_subset_1(c4_4_2_1_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[dh_c4_4_2_1_1__bhsp_7,e7_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,c4_4_2_1_1__bhsp_7),[file(bhsp_7,c4_4_2_1_1__bhsp_7)]]). fof(e8_4_2_1_1__bhsp_7,plain, ( ~ v1_xboole_0(c4_4_2_1_1__bhsp_7) & r1_tarski(c4_4_2_1_1__bhsp_7,c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(c4_4_2_1_1__bhsp_7,A) & r1_xreal_0(c3_4_2_1_1__bhsp_7,k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[dh_c4_4_2_1_1__bhsp_7,e7_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e8_4_2_1_1__bhsp_7),[file(bhsp_7,e8_4_2_1_1__bhsp_7)]]). fof(e10_4_2_1_1__bhsp_7,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( A = c1_4_2_1_1__bhsp_7 => ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(c4_4_2_1_1__bhsp_7,B) & r1_xreal_0(k6_real_1(1,k3_real_1(A,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_nat_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k3_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k6_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_1_1__bhsp_7,dt_c4_4_2_1_1__bhsp_7,de_c3_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e8_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e10_4_2_1_1__bhsp_7),[file(bhsp_7,e10_4_2_1_1__bhsp_7)]]). fof(d1_zfmisc_1,definition,( ! [A,B] : ( B = k1_zfmisc_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> r1_tarski(C,A) ) ) ), file(zfmisc_1,d1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,d1_zfmisc_1)]). fof(e9_4_2_1_1__bhsp_7,plain,( r2_hidden(c4_4_2_1_1__bhsp_7,k1_zfmisc_1(c2_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc2_finset_1,rc4_struct_0,existence_l1_rlvect_1,dt_k5_ordinal2,dt_l1_rlvect_1,fc1_ordinal2,fc5_membered,rc3_funct_1,commutativity_k2_xcmplx_0,existence_l2_rlvect_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k2_xcmplx_0,dt_k3_supinf_1,dt_k5_numbers,dt_k7_xcmplx_0,dt_l2_rlvect_1,dt_m2_subset_1,dt_c1_4_2_1_1__bhsp_7,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc30_xreal_0,fc3_xreal_0,fc6_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t2_real,t3_real,t4_real,t5_real,t6_arithm,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k1_nat_1,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k1_nat_1,redefinition_k6_real_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_nat_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k6_real_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc6_membered,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc1_boole,spc1_numerals,projectivity_k18_complex1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k18_complex1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_1_1__bhsp_7,dt_c4_4_2_1_1__bhsp_7,de_c3_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,e8_4_2_1_1__bhsp_7,d1_zfmisc_1]), [interesting(0.35),file(bhsp_7,e9_4_2_1_1__bhsp_7),[file(bhsp_7,e9_4_2_1_1__bhsp_7)]]). fof(e11_4_2_1_1__bhsp_7,plain,( ? [A] : ( r2_hidden(A,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( m1_subset_1(B,k1_numbers) => ( B = c1_4_2_1_1__bhsp_7 => ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(A,C) & r1_xreal_0(k6_real_1(1,k3_real_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_nat_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c2_4_2_1_1__bhsp_7,de_c2_4_2_1_1__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k3_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k6_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_1_1__bhsp_7,dt_c4_4_2_1_1__bhsp_7,de_c3_4_2_1_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e10_4_2_1_1__bhsp_7,e8_4_2_1_1__bhsp_7,e9_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e11_4_2_1_1__bhsp_7),[file(bhsp_7,e11_4_2_1_1__bhsp_7)]]). fof(i3_4_2_1_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_1_1__bhsp_7)]), [interesting(0.35),trivial,file(bhsp_7,i3_4_2_1_1__bhsp_7)]). fof(i2_4_2_1_1__bhsp_7,plain,( ? [A] : ( r2_hidden(A,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( m1_subset_1(B,k1_numbers) => ( B = c1_4_2_1_1__bhsp_7 => ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(A,C) & r1_xreal_0(k6_real_1(1,k3_real_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e1_4_2_1_1__bhsp_7,e2_4__bhsp_7])],[e11_4_2_1_1__bhsp_7,i3_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,i2_4_2_1_1__bhsp_7),[file(bhsp_7,i2_4_2_1_1__bhsp_7)]]). fof(i1_4_2_1_1__bhsp_7,plain,( ~ ( r2_hidden(c1_4_2_1_1__bhsp_7,k5_numbers) & ! [A] : ~ ( r2_hidden(A,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( m1_subset_1(B,k1_numbers) => ( B = c1_4_2_1_1__bhsp_7 => ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(A,C) & r1_xreal_0(k6_real_1(1,k3_real_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c1_4_2_1_1__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_1_1__bhsp_7])],[e1_4_2_1_1__bhsp_7,i2_4_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,i1_4_2_1_1__bhsp_7),[file(bhsp_7,i1_4_2_1_1__bhsp_7)]]). fof(i1_4_2_1_1_tmp__bhsp_7,plain,( ~ ( r2_hidden(c1_4_2_1_1__bhsp_7,k5_numbers) & ! [A] : ~ ( r2_hidden(A,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & ! [B] : ( m1_subset_1(B,k1_numbers) => ( B = c1_4_2_1_1__bhsp_7 => ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(A,C) & r1_xreal_0(k6_real_1(1,k3_real_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_1_1__bhsp_7])],[dt_c1_4_2_1_1__bhsp_7,i1_4_2_1_1__bhsp_7]), [interesting(0.5),e1_4_2_1__bhsp_7]). fof(e1_4_2_1__bhsp_7,plain,( ! [A] : ~ ( r2_hidden(A,k5_numbers) & ! [B] : ~ ( r2_hidden(B,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( m1_subset_1(C,k1_numbers) => ( C = A => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(B,D) & r1_xreal_0(k6_real_1(1,k3_real_1(C,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[i1_4_2_1_1_tmp__bhsp_7,dh_c1_4_2_1_1__bhsp_7]), [interesting(0.5),file(bhsp_7,e1_4_2_1__bhsp_7),[file(bhsp_7,e1_4_2_1__bhsp_7)]]). fof(e2_4_2_1__bhsp_7,plain,( ! [A] : ~ ( r2_hidden(A,k5_numbers) & ! [B] : ~ ( r2_hidden(B,k1_zfmisc_1(c2_4__bhsp_7)) & v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & ! [C] : ( m1_subset_1(C,k1_numbers) => ( C = A => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(B,D) & r1_xreal_0(k6_real_1(1,k3_real_1(C,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k6_real_1,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e1_4_2_1__bhsp_7]), [interesting(0.5),file(bhsp_7,e2_4_2_1__bhsp_7),[file(bhsp_7,e2_4_2_1__bhsp_7)]]). fof(e3_4_2_1__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( r2_hidden(B,k5_numbers) => ( v1_finset_1(k1_funct_1(A,B)) & m1_subset_1(k1_funct_1(A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(A,B)) & r1_tarski(k1_funct_1(A,B),c2_4__bhsp_7) & ! [C] : ( m1_subset_1(C,k1_numbers) => ( C = B => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(A,B),D) & r1_xreal_0(k6_real_1(1,k3_real_1(C,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7))) ) ) ) ) ) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k2_xcmplx_0,redefinition_m2_subset_1,dt_k16_complex1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_k7_xcmplx_0,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k18_complex1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,spc1_numerals,s1_funct_2__e3_4_2_1__bhsp_7,e2_4_2_1__bhsp_7]), [interesting(0.5),file(bhsp_7,e3_4_2_1__bhsp_7),[file(bhsp_7,e3_4_2_1__bhsp_7)]]). fof(i1_4_2_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i1_4_2_1__bhsp_7)]), [interesting(0.5),trivial,file(bhsp_7,i1_4_2_1__bhsp_7)]). fof(e1_4_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( r2_hidden(B,k5_numbers) => ( v1_finset_1(k1_funct_1(A,B)) & m1_subset_1(k1_funct_1(A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(A,B)) & r1_tarski(k1_funct_1(A,B),c2_4__bhsp_7) & ! [C] : ( m1_subset_1(C,k1_numbers) => ( C = B => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(A,B),D) & r1_xreal_0(k6_real_1(1,k3_real_1(C,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7))) ) ) ) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[e3_4_2_1__bhsp_7,i1_4_2_1__bhsp_7]), [interesting(0.65),file(bhsp_7,e1_4_2__bhsp_7),[file(bhsp_7,e1_4_2__bhsp_7)]]). fof(e1_4_2_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( r2_hidden(B,k5_numbers) => ( v1_finset_1(k1_funct_1(A,B)) & m1_subset_1(k1_funct_1(A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(A,B)) & r1_tarski(k1_funct_1(A,B),c2_4__bhsp_7) & ! [C] : ( m1_subset_1(C,k1_numbers) => ( C = B => ! [D] : ( ( v1_finset_1(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(D) & r1_tarski(D,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(A,B),D) & r1_xreal_0(k6_real_1(1,k3_real_1(C,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,D,c3_4__bhsp_7))) ) ) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k18_complex1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_relset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e1_4_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e1_4_2_2__bhsp_7),[file(bhsp_7,e1_4_2_2__bhsp_7)]]). fof(dt_c1_4_2_2__bhsp_7,plain, ( v1_funct_1(c1_4_2_2__bhsp_7) & v1_funct_2(c1_4_2_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(c1_4_2_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c1_4_2_2__bhsp_7,e1_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,c1_4_2_2__bhsp_7),[file(bhsp_7,c1_4_2_2__bhsp_7)]]). fof(s3_recdef_1__e3_4_2_2__bhsp_7,theorem,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_zfmisc_1(B)) & m2_relset_1(C,k5_numbers,k1_zfmisc_1(B)) ) => ? [D] : ( v1_relat_1(D) & v1_funct_1(D) & k1_relat_1(D) = k5_numbers & k1_funct_1(D,0) = k8_funct_2(k5_numbers,k1_zfmisc_1(B),C,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k1_funct_1(D,k1_nat_1(E,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(B),C,k1_nat_1(E,1)),k1_funct_1(D,E)) ) ) ) ), file(bhsp_7,s3_recdef_1__e3_4_2_2__bhsp_7), [interesting(0.9),axiom,file(bhsp_7,s3_recdef_1__e3_4_2_2__bhsp_7)]). fof(e3_4_2_2__bhsp_7,plain,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = k5_numbers & k1_funct_1(A,0) = k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_funct_1(A,k1_nat_1(B,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(B,1)),k1_funct_1(A,B)) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[dt_l2_struct_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,dt_l1_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc3_xreal_0,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_xreal_0,rc2_finset_1,rc3_finset_1,rc3_funct_1,rc4_finset_1,commutativity_k2_xcmplx_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc1_ordinal2,fc1_struct_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_2,rc1_membered,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4_2_2__bhsp_7,dt_c2_4__bhsp_7,cc6_membered,cc9_membered,fc2_membered,rc1_funct_1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,s3_recdef_1__e3_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e3_4_2_2__bhsp_7),[file(bhsp_7,e3_4_2_2__bhsp_7)]]). fof(e4_4_2_2__bhsp_7,plain,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = k5_numbers & k1_funct_1(A,0) = k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_funct_1(A,k1_nat_1(B,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(B,1)),k1_funct_1(A,B)) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c2_4__bhsp_7,fc2_membered,rc1_funct_1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e3_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e4_4_2_2__bhsp_7),[file(bhsp_7,e4_4_2_2__bhsp_7)]]). fof(dt_c2_4_2_2__bhsp_7,plain, ( v1_relat_1(c2_4_2_2__bhsp_7) & v1_funct_1(c2_4_2_2__bhsp_7) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[dh_c2_4_2_2__bhsp_7,e4_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,c2_4_2_2__bhsp_7),[file(bhsp_7,c2_4_2_2__bhsp_7)]]). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), [interesting(0.9),axiom,file(relat_1,k2_relat_1)]). fof(dt_c1_4_2_2_3__bhsp_7,assumption,( $true ), introduced(assumption,[file(bhsp_7,c1_4_2_2_3__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_2_3__bhsp_7)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(dh_c1_4_2_2_3__bhsp_7,definition, ( ~ ( r2_hidden(c1_4_2_2_3__bhsp_7,k2_relat_1(c2_4_2_2__bhsp_7)) & ~ r2_hidden(c1_4_2_2_3__bhsp_7,k1_zfmisc_1(c2_4__bhsp_7)) ) => ! [A] : ~ ( r2_hidden(A,k2_relat_1(c2_4_2_2__bhsp_7)) & ~ r2_hidden(A,k1_zfmisc_1(c2_4__bhsp_7)) ) ), introduced(definition,[new_symbol(c1_4_2_2_3__bhsp_7),file(bhsp_7,c1_4_2_2_3__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_2_3__bhsp_7)]). fof(e1_4_2_2_3__bhsp_7,assumption,( r2_hidden(c1_4_2_2_3__bhsp_7,k2_relat_1(c2_4_2_2__bhsp_7)) ), introduced(assumption,[file(bhsp_7,e1_4_2_2_3__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,e1_4_2_2_3__bhsp_7)]). fof(dh_c2_4_2_2_3__bhsp_7,definition, ( ? [A] : ( r2_hidden(A,k1_relat_1(c2_4_2_2__bhsp_7)) & c1_4_2_2_3__bhsp_7 = k1_funct_1(c2_4_2_2__bhsp_7,A) ) => ( r2_hidden(c2_4_2_2_3__bhsp_7,k1_relat_1(c2_4_2_2__bhsp_7)) & c1_4_2_2_3__bhsp_7 = k1_funct_1(c2_4_2_2__bhsp_7,c2_4_2_2_3__bhsp_7) ) ), introduced(definition,[new_symbol(c2_4_2_2_3__bhsp_7),file(bhsp_7,c2_4_2_2_3__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c2_4_2_2_3__bhsp_7)]). fof(d5_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), [interesting(0.9),axiom,file(funct_1,d5_funct_1)]). fof(e2_4_2_2_3__bhsp_7,plain,( ? [A] : ( r2_hidden(A,k1_relat_1(c2_4_2_2__bhsp_7)) & c1_4_2_2_3__bhsp_7 = k1_funct_1(c2_4_2_2__bhsp_7,A) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc3_funct_1,existence_m1_subset_1,dt_m1_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k2_relat_1,dt_c1_4_2_2_3__bhsp_7,dt_c2_4_2_2__bhsp_7,rc1_funct_1,t1_subset,t7_boole,e1_4_2_2_3__bhsp_7,d5_funct_1]), [interesting(0.35),file(bhsp_7,e2_4_2_2_3__bhsp_7),[file(bhsp_7,e2_4_2_2_3__bhsp_7)]]). fof(dt_c2_4_2_2_3__bhsp_7,plain,( $true ), inference(consider,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[dh_c2_4_2_2_3__bhsp_7,e2_4_2_2_3__bhsp_7]), [interesting(0.35),file(bhsp_7,c2_4_2_2_3__bhsp_7),[file(bhsp_7,c2_4_2_2_3__bhsp_7)]]). fof(de_c3_4_2_2_3__bhsp_7,definition,( c3_4_2_2_3__bhsp_7 = c2_4_2_2_3__bhsp_7 ), introduced(definition,[new_symbol(c3_4_2_2_3__bhsp_7),file(bhsp_7,c3_4_2_2_3__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c3_4_2_2_3__bhsp_7)]). fof(e5_4_2_2__bhsp_7,plain, ( k1_relat_1(c2_4_2_2__bhsp_7) = k5_numbers & k1_funct_1(c2_4_2_2__bhsp_7,0) = k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(A,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,commutativity_k2_xcmplx_0,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,rc1_funct_1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,dh_c2_4_2_2__bhsp_7,e4_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e5_4_2_2__bhsp_7),[file(bhsp_7,e5_4_2_2__bhsp_7)]]). fof(e3_4_2_2_3__bhsp_7,plain, ( r2_hidden(c2_4_2_2_3__bhsp_7,k1_relat_1(c2_4_2_2__bhsp_7)) & c1_4_2_2_3__bhsp_7 = k1_funct_1(c2_4_2_2__bhsp_7,c2_4_2_2_3__bhsp_7) ), inference(consider,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[dh_c2_4_2_2_3__bhsp_7,e2_4_2_2_3__bhsp_7]), [interesting(0.35),file(bhsp_7,e3_4_2_2_3__bhsp_7),[file(bhsp_7,e3_4_2_2_3__bhsp_7)]]). fof(e4_4_2_2_3__bhsp_7,plain,( m2_subset_1(c2_4_2_2_3__bhsp_7,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_3__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c2_4_2_2_3__bhsp_7,fc2_membered,t1_subset,t7_boole,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e5_4_2_2__bhsp_7,e3_4_2_2_3__bhsp_7]), [interesting(0.35),file(bhsp_7,e4_4_2_2_3__bhsp_7),[file(bhsp_7,e4_4_2_2_3__bhsp_7)]]). fof(dt_c3_4_2_2_3__bhsp_7,plain,( m2_subset_1(c3_4_2_2_3__bhsp_7,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c2_4_2_2_3__bhsp_7,fc2_membered,de_c3_4_2_2_3__bhsp_7,e4_4_2_2_3__bhsp_7]), [interesting(0.35),file(bhsp_7,c3_4_2_2_3__bhsp_7),[file(bhsp_7,c3_4_2_2_3__bhsp_7)]]). fof(s1_nat_1__e9_4_2_2__bhsp_7,theorem,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k6_supinf_1) & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) & v1_relat_1(D) & v1_funct_1(D) ) => ( ( v1_finset_1(k1_funct_1(D,0)) & m1_subset_1(k1_funct_1(D,0),k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(k1_funct_1(D,0)) & r1_tarski(k1_funct_1(D,0),B) & ! [E] : ( ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(E) & r1_tarski(E,B) & r1_xboole_0(k1_funct_1(D,0),E) & r1_xreal_0(k6_real_1(1,k1_nat_1(0,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,E,C))) ) ) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_xreal_0(0,E) => r1_tarski(k1_funct_1(D,0),k1_funct_1(D,E)) ) ) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( ( v1_finset_1(k1_funct_1(D,F)) & m1_subset_1(k1_funct_1(D,F),k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(k1_funct_1(D,F)) & r1_tarski(k1_funct_1(D,F),B) & ! [G] : ( ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(G) & r1_tarski(G,B) & r1_xboole_0(k1_funct_1(D,F),G) & r1_xreal_0(k6_real_1(1,k1_nat_1(F,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C))) ) ) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( r1_xreal_0(F,G) => r1_tarski(k1_funct_1(D,F),k1_funct_1(D,G)) ) ) ) => ( v1_finset_1(k1_funct_1(D,k1_nat_1(F,1))) & m1_subset_1(k1_funct_1(D,k1_nat_1(F,1)),k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(k1_funct_1(D,k1_nat_1(F,1))) & r1_tarski(k1_funct_1(D,k1_nat_1(F,1)),B) & ! [H] : ( ( v1_finset_1(H) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(H) & r1_tarski(H,B) & r1_xboole_0(k1_funct_1(D,k1_nat_1(F,1)),H) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(F,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,H,C))) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(F,1),H) => r1_tarski(k1_funct_1(D,k1_nat_1(F,1)),k1_funct_1(D,H)) ) ) ) ) ) ) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( v1_finset_1(k1_funct_1(D,F)) & m1_subset_1(k1_funct_1(D,F),k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(k1_funct_1(D,F)) & r1_tarski(k1_funct_1(D,F),B) & ! [I] : ( ( v1_finset_1(I) & m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ v1_xboole_0(I) & r1_tarski(I,B) & r1_xboole_0(k1_funct_1(D,F),I) & r1_xreal_0(k6_real_1(1,k1_nat_1(F,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,I,C))) ) ) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( r1_xreal_0(F,I) => r1_tarski(k1_funct_1(D,F),k1_funct_1(D,I)) ) ) ) ) ) ) ), file(bhsp_7,s1_nat_1__e9_4_2_2__bhsp_7), [interesting(0.9),axiom,file(bhsp_7,s1_nat_1__e9_4_2_2__bhsp_7)]). fof(spc5_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ), file(arithm,spc5_arithm), [interesting(0.9),axiom,file(arithm,spc5_arithm)]). fof(dh_c1_4_2_2_1__bhsp_7,definition, ( ( m2_subset_1(c1_4_2_2_1__bhsp_7,k1_numbers,k5_numbers) => ( c1_4_2_2_1__bhsp_7 = 0 => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = 0 => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,B),k1_funct_1(c2_4_2_2__bhsp_7,C)) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_2_1__bhsp_7),file(bhsp_7,c1_4_2_2_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_2_1__bhsp_7)]). fof(e1_4_2_2_1__bhsp_7,assumption,( c1_4_2_2_1__bhsp_7 = 0 ), introduced(assumption,[file(bhsp_7,e1_4_2_2_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,e1_4_2_2_1__bhsp_7)]). fof(dt_c1_4_2_2_1__bhsp_7,assumption,( m2_subset_1(c1_4_2_2_1__bhsp_7,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_7,c1_4_2_2_1__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_2_1__bhsp_7)]). fof(s1_nat_1__e4_4_2_2_1__bhsp_7,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( ( ( r1_xreal_0(B,0) => r1_tarski(k1_funct_1(A,B),k1_funct_1(A,0)) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(B,C) => r1_tarski(k1_funct_1(A,B),k1_funct_1(A,C)) ) => ( r1_xreal_0(B,k1_nat_1(C,1)) => r1_tarski(k1_funct_1(A,B),k1_funct_1(A,k1_nat_1(C,1))) ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k1_funct_1(A,B),k1_funct_1(A,C)) ) ) ) ) ), file(bhsp_7,s1_nat_1__e4_4_2_2_1__bhsp_7), [interesting(0.9),axiom,file(bhsp_7,s1_nat_1__e4_4_2_2_1__bhsp_7)]). fof(e2_4_2_2_1__bhsp_7,plain, ( r1_xreal_0(c1_4_2_2_1__bhsp_7,0) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7])],[cc1_xreal_0,rc2_finset_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc2_membered,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k1_funct_1,dt_c1_4_2_2_1__bhsp_7,dt_c2_4_2_2__bhsp_7,rqLessOrEqual__r1_xreal_0__r0_r0,t3_subset,spc0_boole,spc0_numerals,e1_4_2_2_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e2_4_2_2_1__bhsp_7),[file(bhsp_7,e2_4_2_2_1__bhsp_7)]]). fof(dh_c1_4_2_2_1_1__bhsp_7,definition, ( ( m2_subset_1(c1_4_2_2_1_1__bhsp_7,k1_numbers,k5_numbers) => ( ( r1_xreal_0(c1_4_2_2_1__bhsp_7,c1_4_2_2_1_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1_1__bhsp_7)) ) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1))) ) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(A,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1))) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_2_1_1__bhsp_7),file(bhsp_7,c1_4_2_2_1_1__bhsp_7)]), [interesting(0.2),axiom,file(bhsp_7,c1_4_2_2_1_1__bhsp_7)]). fof(e1_4_2_2_1_1__bhsp_7,assumption, ( r1_xreal_0(c1_4_2_2_1__bhsp_7,c1_4_2_2_1_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1_1__bhsp_7)) ), introduced(assumption,[file(bhsp_7,e1_4_2_2_1_1__bhsp_7)]), [interesting(0.2),axiom,file(bhsp_7,e1_4_2_2_1_1__bhsp_7)]). fof(dt_c1_4_2_2_1_1__bhsp_7,assumption,( m2_subset_1(c1_4_2_2_1_1__bhsp_7,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_7,c1_4_2_2_1_1__bhsp_7)]), [interesting(0.2),axiom,file(bhsp_7,c1_4_2_2_1_1__bhsp_7)]). fof(e2_4_2_2_1_1__bhsp_7,plain,( k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1_1__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e5_4_2_2__bhsp_7]), [interesting(0.2),file(bhsp_7,e2_4_2_2_1_1__bhsp_7),[file(bhsp_7,e2_4_2_2_1_1__bhsp_7)]]). fof(e3_4_2_2_1_1__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc2_finset_1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc2_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_1_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,t3_subset,spc1_boole,spc1_numerals,e2_4_2_2_1_1__bhsp_7,t7_xboole_1]), [interesting(0.2),file(bhsp_7,e3_4_2_2_1_1__bhsp_7),[file(bhsp_7,e3_4_2_2_1_1__bhsp_7)]]). fof(t1_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), [interesting(0.9),axiom,file(xboole_1,t1_xboole_1)]). fof(e4_4_2_2_1_1__bhsp_7,plain, ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4_2_2_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7,e1_4_2_2_1_1__bhsp_7])],[rc2_finset_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_arithm,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_funct_1,dt_k1_nat_1,dt_c1_4_2_2_1__bhsp_7,dt_c1_4_2_2_1_1__bhsp_7,dt_c2_4_2_2__bhsp_7,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,t3_subset,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e3_4_2_2_1_1__bhsp_7,e1_4_2_2_1__bhsp_7,e1_4_2_2_1_1__bhsp_7,t18_nat_1,t1_xboole_1]), [interesting(0.2),file(bhsp_7,e4_4_2_2_1_1__bhsp_7),[file(bhsp_7,e4_4_2_2_1_1__bhsp_7)]]). fof(i3_4_2_2_1_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_2_1_1__bhsp_7)]), [interesting(0.2),trivial,file(bhsp_7,i3_4_2_2_1_1__bhsp_7)]). fof(i2_4_2_2_1_1__bhsp_7,plain, ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4_2_2_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7,e1_4_2_2_1_1__bhsp_7])],[e4_4_2_2_1_1__bhsp_7,i3_4_2_2_1_1__bhsp_7]), [interesting(0.2),file(bhsp_7,i2_4_2_2_1_1__bhsp_7),[file(bhsp_7,i2_4_2_2_1_1__bhsp_7)]]). fof(i1_4_2_2_1_1__bhsp_7,plain, ( ( r1_xreal_0(c1_4_2_2_1__bhsp_7,c1_4_2_2_1_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1_1__bhsp_7)) ) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4_2_2_1_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_1_1__bhsp_7])],[e1_4_2_2_1_1__bhsp_7,i2_4_2_2_1_1__bhsp_7]), [interesting(0.2),file(bhsp_7,i1_4_2_2_1_1__bhsp_7),[file(bhsp_7,i1_4_2_2_1_1__bhsp_7)]]). fof(i1_4_2_2_1_1_tmp__bhsp_7,plain, ( m2_subset_1(c1_4_2_2_1_1__bhsp_7,k1_numbers,k5_numbers) => ( ( r1_xreal_0(c1_4_2_2_1__bhsp_7,c1_4_2_2_1_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1_1__bhsp_7)) ) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_1_1__bhsp_7,1))) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_2_1_1__bhsp_7])],[dt_c1_4_2_2_1_1__bhsp_7,i1_4_2_2_1_1__bhsp_7]), [interesting(0.35),e3_4_2_2_1__bhsp_7]). fof(e3_4_2_2_1__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,k1_nat_1(A,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1))) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7])],[i1_4_2_2_1_1_tmp__bhsp_7,dh_c1_4_2_2_1_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e3_4_2_2_1__bhsp_7),[file(bhsp_7,e3_4_2_2_1__bhsp_7)]]). fof(e4_4_2_2_1__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7])],[cc1_xreal_0,rc2_finset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,commutativity_k2_xcmplx_0,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_4_2_2_1__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,rc1_funct_1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,s1_nat_1__e4_4_2_2_1__bhsp_7,e2_4_2_2_1__bhsp_7,e3_4_2_2_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e4_4_2_2_1__bhsp_7),[file(bhsp_7,e4_4_2_2_1__bhsp_7)]]). fof(i3_4_2_2_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_2_1__bhsp_7)]), [interesting(0.35),trivial,file(bhsp_7,i3_4_2_2_1__bhsp_7)]). fof(i2_4_2_2_1__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_1__bhsp_7])],[e4_4_2_2_1__bhsp_7,i3_4_2_2_1__bhsp_7]), [interesting(0.35),file(bhsp_7,i2_4_2_2_1__bhsp_7),[file(bhsp_7,i2_4_2_2_1__bhsp_7)]]). fof(i1_4_2_2_1__bhsp_7,plain, ( c1_4_2_2_1__bhsp_7 = 0 => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_1__bhsp_7])],[e1_4_2_2_1__bhsp_7,i2_4_2_2_1__bhsp_7]), [interesting(0.35),file(bhsp_7,i1_4_2_2_1__bhsp_7),[file(bhsp_7,i1_4_2_2_1__bhsp_7)]]). fof(i1_4_2_2_1_tmp__bhsp_7,plain, ( m2_subset_1(c1_4_2_2_1__bhsp_7,k1_numbers,k5_numbers) => ( c1_4_2_2_1__bhsp_7 = 0 => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_1__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_2_1__bhsp_7])],[dt_c1_4_2_2_1__bhsp_7,i1_4_2_2_1__bhsp_7]), [interesting(0.5),e6_4_2_2__bhsp_7]). fof(e6_4_2_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( A = 0 => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,A),k1_funct_1(c2_4_2_2__bhsp_7,B)) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[i1_4_2_2_1_tmp__bhsp_7,dh_c1_4_2_2_1__bhsp_7]), [interesting(0.5),file(bhsp_7,e6_4_2_2__bhsp_7),[file(bhsp_7,e6_4_2_2__bhsp_7)]]). fof(e2_4_2_2__bhsp_7,plain,( ! [A] : ( r2_hidden(A,k5_numbers) => ( v1_finset_1(k1_funct_1(c1_4_2_2__bhsp_7,A)) & m1_subset_1(k1_funct_1(c1_4_2_2__bhsp_7,A),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c1_4_2_2__bhsp_7,A)) & r1_tarski(k1_funct_1(c1_4_2_2__bhsp_7,A),c2_4__bhsp_7) & ! [B] : ( m1_subset_1(B,k1_numbers) => ( B = A => ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c1_4_2_2__bhsp_7,A),C) & r1_xreal_0(k6_real_1(1,k3_real_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) ) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c1_4_2_2__bhsp_7,e1_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e2_4_2_2__bhsp_7),[file(bhsp_7,e2_4_2_2__bhsp_7)]]). fof(e7_4_2_2__bhsp_7,plain, ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,0)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,0),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,0)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,0),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,0),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(0,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(0,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,0),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc23_xreal_0,fc24_membered,fc25_membered,fc26_membered,fc30_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_boole,t1_real,t2_arithm,t3_arithm,t4_real,t5_arithm,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k33_binop_2,dt_k3_real_1,dt_k3_xcmplx_0,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k7_xcmplx_0,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc2_xboole_0,fc3_binop_2,fc3_xboole_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc0_boole,spc1_boole,t1_numerals,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e6_4_2_2__bhsp_7,e2_4_2_2__bhsp_7,e5_4_2_2__bhsp_7,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_7,e7_4_2_2__bhsp_7),[file(bhsp_7,e7_4_2_2__bhsp_7)]]). fof(dh_c1_4_2_2_2__bhsp_7,definition, ( ( m2_subset_1(c1_4_2_2_2__bhsp_7,k1_numbers,k5_numbers) => ( ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_2__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,B)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,B)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,B),c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,B),C) & r1_xreal_0(k6_real_1(1,k1_nat_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,B),k1_funct_1(c2_4_2_2__bhsp_7,C)) ) ) ) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(B,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(B,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(B,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(B,1)),c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(B,1)),C) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(B,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(B,1),C) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(B,1)),k1_funct_1(c2_4_2_2__bhsp_7,C)) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_2_2__bhsp_7),file(bhsp_7,c1_4_2_2_2__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_2_2__bhsp_7)]). fof(dt_c1_4_2_2_2__bhsp_7,assumption,( m2_subset_1(c1_4_2_2_2__bhsp_7,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_7,c1_4_2_2_2__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,c1_4_2_2_2__bhsp_7)]). fof(e1_4_2_2_2__bhsp_7,assumption, ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_2__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), introduced(assumption,[file(bhsp_7,e1_4_2_2_2__bhsp_7)]), [interesting(0.35),axiom,file(bhsp_7,e1_4_2_2_2__bhsp_7)]). fof(rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1,theorem,( k2_xcmplx_0(k7_xcmplx_0(1,2),k7_xcmplx_0(1,2)) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1)]). fof(s1_nat_1__e3_4_2_2_2_2__bhsp_7,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( ( ( r1_xreal_0(k1_nat_1(B,1),0) => r1_tarski(k1_funct_1(A,k1_nat_1(B,1)),k1_funct_1(A,0)) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(k1_nat_1(B,1),C) => r1_tarski(k1_funct_1(A,k1_nat_1(B,1)),k1_funct_1(A,C)) ) => ( r1_xreal_0(k1_nat_1(B,1),k1_nat_1(C,1)) => r1_tarski(k1_funct_1(A,k1_nat_1(B,1)),k1_funct_1(A,k1_nat_1(C,1))) ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(B,1),C) => r1_tarski(k1_funct_1(A,k1_nat_1(B,1)),k1_funct_1(A,C)) ) ) ) ) ), file(bhsp_7,s1_nat_1__e3_4_2_2_2_2__bhsp_7), [interesting(0.9),axiom,file(bhsp_7,s1_nat_1__e3_4_2_2_2_2__bhsp_7)]). fof(t19_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ~ ( 0 != A & r1_xreal_0(A,0) ) ) ), file(nat_1,t19_nat_1), [interesting(0.9),axiom,file(nat_1,t19_nat_1)]). fof(e1_4_2_2_2_2__bhsp_7,plain, ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),0) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[rc2_finset_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_arithm,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_funct_1,dt_k1_nat_1,dt_c1_4_2_2_2__bhsp_7,dt_c2_4_2_2__bhsp_7,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r0,t3_subset,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,t19_nat_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(bhsp_7,e1_4_2_2_2_2__bhsp_7),[file(bhsp_7,e1_4_2_2_2_2__bhsp_7)]]). fof(dh_c1_4_2_2_2_2_1__bhsp_7,definition, ( ( m2_subset_1(c1_4_2_2_2_2_1__bhsp_7,k1_numbers,k5_numbers) => ( ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),c1_4_2_2_2_2_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7)) ) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),k1_nat_1(A,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1))) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_2_2_2_1__bhsp_7),file(bhsp_7,c1_4_2_2_2_2_1__bhsp_7)]), [interesting(0.05),axiom,file(bhsp_7,c1_4_2_2_2_2_1__bhsp_7)]). fof(e1_4_2_2_2_2_1__bhsp_7,assumption, ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),c1_4_2_2_2_2_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7)) ), introduced(assumption,[file(bhsp_7,e1_4_2_2_2_2_1__bhsp_7)]), [interesting(0.05),axiom,file(bhsp_7,e1_4_2_2_2_2_1__bhsp_7)]). fof(e2_4_2_2_2_2_1__bhsp_7,assumption,( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1)) ), introduced(assumption,[file(bhsp_7,e2_4_2_2_2_2_1__bhsp_7)]), [interesting(0.05),axiom,file(bhsp_7,e2_4_2_2_2_2_1__bhsp_7)]). fof(dt_c1_4_2_2_2_2_1__bhsp_7,assumption,( m2_subset_1(c1_4_2_2_2_2_1__bhsp_7,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_7,c1_4_2_2_2_2_1__bhsp_7)]), [interesting(0.05),axiom,file(bhsp_7,c1_4_2_2_2_2_1__bhsp_7)]). fof(e1_4_2_2_2_2_1_1_1_1__bhsp_7,assumption,( c1_4_2_2_2__bhsp_7 = c1_4_2_2_2_2_1__bhsp_7 ), introduced(assumption,[file(bhsp_7,e1_4_2_2_2_2_1_1_1_1__bhsp_7)]), [interesting(0.02),axiom,file(bhsp_7,e1_4_2_2_2_2_1_1_1_1__bhsp_7)]). fof(e2_4_2_2_2_2_1_1_1_1__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1_1_1_1__bhsp_7])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc2_finset_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc4_finset_1,spc6_arithm,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc6_membered,cc9_membered,fc2_membered,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,redefinition_k1_nat_1,dt_k1_funct_1,dt_k1_nat_1,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c2_4_2_2__bhsp_7,t3_subset,spc1_boole,spc1_numerals,e1_4_2_2_2_2_1_1_1_1__bhsp_7]), [interesting(0.02),file(bhsp_7,e2_4_2_2_2_2_1_1_1_1__bhsp_7),[file(bhsp_7,e2_4_2_2_2_2_1_1_1_1__bhsp_7)]]). fof(i2_4_2_2_2_2_1_1_1_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i2_4_2_2_2_2_1_1_1_1__bhsp_7)]), [interesting(0.02),trivial,file(bhsp_7,i2_4_2_2_2_2_1_1_1_1__bhsp_7)]). fof(i1_4_2_2_2_2_1_1_1_1__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1_1_1_1__bhsp_7])],[e2_4_2_2_2_2_1_1_1_1__bhsp_7,i2_4_2_2_2_2_1_1_1_1__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_2_2_2_1_1_1_1__bhsp_7),[file(bhsp_7,i1_4_2_2_2_2_1_1_1_1__bhsp_7)]]). fof(i1_4_2_2_2_2_1_1_1__bhsp_7,plain, ( c1_4_2_2_2__bhsp_7 = c1_4_2_2_2_2_1__bhsp_7 => ( c1_4_2_2_2__bhsp_7 = c1_4_2_2_2_2_1__bhsp_7 & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_2_2_1_1_1_1__bhsp_7])],[e1_4_2_2_2_2_1_1_1_1__bhsp_7,i1_4_2_2_2_2_1_1_1_1__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_2_2_2_1_1_1__bhsp_7),[file(bhsp_7,i1_4_2_2_2_2_1_1_1__bhsp_7)]]). fof(e1_4_2_2_2_2_1_1_1_2__bhsp_7,assumption,( c1_4_2_2_2__bhsp_7 != c1_4_2_2_2_2_1__bhsp_7 ), introduced(assumption,[file(bhsp_7,e1_4_2_2_2_2_1_1_1_2__bhsp_7)]), [interesting(0.02),axiom,file(bhsp_7,e1_4_2_2_2_2_1_1_1_2__bhsp_7)]). fof(e4_4_2_2_2_2_1_1_1_2__bhsp_7,plain,( k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e5_4_2_2__bhsp_7]), [interesting(0.02),file(bhsp_7,e4_4_2_2_2_2_1_1_1_2__bhsp_7),[file(bhsp_7,e4_4_2_2_2_2_1_1_1_2__bhsp_7)]]). fof(e5_4_2_2_2_2_1_1_1_2__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc2_finset_1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc2_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,t3_subset,spc1_boole,spc1_numerals,e4_4_2_2_2_2_1_1_1_2__bhsp_7,t7_xboole_1]), [interesting(0.02),file(bhsp_7,e5_4_2_2_2_2_1_1_1_2__bhsp_7),[file(bhsp_7,e5_4_2_2_2_2_1_1_1_2__bhsp_7)]]). fof(t8_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), [interesting(0.9),axiom,file(xreal_1,t8_xreal_1)]). fof(e2_4_2_2_2_2_1_1_1_2__bhsp_7,plain,( r1_xreal_0(c1_4_2_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc8_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,cc2_xreal_0,fc3_xreal_0,spc1_boole,spc1_numerals,e2_4_2_2_2_2_1__bhsp_7,t8_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.02),file(bhsp_7,e2_4_2_2_2_2_1_1_1_2__bhsp_7),[file(bhsp_7,e2_4_2_2_2_2_1_1_1_2__bhsp_7)]]). fof(d5_real_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), [interesting(0.9),axiom,file(real_1,d5_real_1)]). fof(e3_4_2_2_2_2_1_1_1_2__bhsp_7,plain,( ~ r1_xreal_0(c1_4_2_2_2_2_1__bhsp_7,c1_4_2_2_2__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,e1_4_2_2_2_2_1_1_1_2__bhsp_7])],[cc1_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,fc2_membered,rc1_xreal_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,cc2_xreal_0,e2_4_2_2_2_2_1_1_1_2__bhsp_7,e1_4_2_2_2_2_1_1_1_2__bhsp_7,d5_real_1]), [interesting(0.02),file(bhsp_7,e3_4_2_2_2_2_1_1_1_2__bhsp_7),[file(bhsp_7,e3_4_2_2_2_2_1_1_1_2__bhsp_7)]]). fof(e6_4_2_2_2_2_1_1_1_2__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,e1_4_2_2_2_2_1_1_1_2__bhsp_7])],[rc2_finset_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_xboole_0,rc1_xreal_0,rc2_funct_1,rc2_xboole_0,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_funct_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c2_4_2_2__bhsp_7,cc1_xreal_0,t3_subset,spc1_boole,spc1_numerals,e5_4_2_2_2_2_1_1_1_2__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,e3_4_2_2_2_2_1_1_1_2__bhsp_7,t38_nat_1,t1_xboole_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.02),file(bhsp_7,e6_4_2_2_2_2_1_1_1_2__bhsp_7),[file(bhsp_7,e6_4_2_2_2_2_1_1_1_2__bhsp_7)]]). fof(i2_4_2_2_2_2_1_1_1_2__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i2_4_2_2_2_2_1_1_1_2__bhsp_7)]), [interesting(0.02),trivial,file(bhsp_7,i2_4_2_2_2_2_1_1_1_2__bhsp_7)]). fof(i1_4_2_2_2_2_1_1_1_2__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,e1_4_2_2_2_2_1_1_1_2__bhsp_7])],[e6_4_2_2_2_2_1_1_1_2__bhsp_7,i2_4_2_2_2_2_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_2_2_2_1_1_1_2__bhsp_7),[file(bhsp_7,i1_4_2_2_2_2_1_1_1_2__bhsp_7)]]). fof(i2_4_2_2_2_2_1_1_1__bhsp_7,plain, ( c1_4_2_2_2__bhsp_7 != c1_4_2_2_2_2_1__bhsp_7 => ( c1_4_2_2_2__bhsp_7 != c1_4_2_2_2_2_1__bhsp_7 & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_2_2_1_1_1_2__bhsp_7])],[e1_4_2_2_2_2_1_1_1_2__bhsp_7,i1_4_2_2_2_2_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,i2_4_2_2_2_2_1_1_1__bhsp_7),[file(bhsp_7,i2_4_2_2_2_2_1_1_1__bhsp_7)]]). fof(e1_4_2_2_2_2_1_1_1__bhsp_7,plain,( ~ ( c1_4_2_2_2__bhsp_7 != c1_4_2_2_2_2_1__bhsp_7 & c1_4_2_2_2__bhsp_7 = c1_4_2_2_2_2_1__bhsp_7 ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7])],[cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,fc2_membered,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7]), [interesting(0.02),file(bhsp_7,e1_4_2_2_2_2_1_1_1__bhsp_7),[file(bhsp_7,e1_4_2_2_2_2_1_1_1__bhsp_7)]]). fof(e3_4_2_2_2_2_1__bhsp_7,plain, ( ( c1_4_2_2_2__bhsp_7 = c1_4_2_2_2_2_1__bhsp_7 & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) | ( c1_4_2_2_2__bhsp_7 != c1_4_2_2_2_2_1__bhsp_7 & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) ), inference(percases,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7])],[i1_4_2_2_2_2_1_1_1__bhsp_7,i2_4_2_2_2_2_1_1_1__bhsp_7,e1_4_2_2_2_2_1_1_1__bhsp_7]), [interesting(0.05),file(bhsp_7,e3_4_2_2_2_2_1__bhsp_7),[file(bhsp_7,e3_4_2_2_2_2_1__bhsp_7)]]). fof(e4_4_2_2_2_2_1__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc2_finset_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,fc1_ordinal2,fc1_xboole_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc4_finset_1,spc6_arithm,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc6_membered,cc9_membered,fc2_membered,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,redefinition_k1_nat_1,dt_k1_funct_1,dt_k1_nat_1,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7,dt_c2_4_2_2__bhsp_7,t3_subset,spc1_boole,spc1_numerals,e3_4_2_2_2_2_1__bhsp_7]), [interesting(0.05),file(bhsp_7,e4_4_2_2_2_2_1__bhsp_7),[file(bhsp_7,e4_4_2_2_2_2_1__bhsp_7)]]). fof(i3_4_2_2_2_2_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_2_2_2_1__bhsp_7)]), [interesting(0.05),trivial,file(bhsp_7,i3_4_2_2_2_2_1__bhsp_7)]). fof(i2_4_2_2_2_2_1__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7])],[e4_4_2_2_2_2_1__bhsp_7,i3_4_2_2_2_2_1__bhsp_7]), [interesting(0.05),file(bhsp_7,i2_4_2_2_2_2_1__bhsp_7),[file(bhsp_7,i2_4_2_2_2_2_1__bhsp_7)]]). fof(i1_4_2_2_2_2_1__bhsp_7,plain, ( ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),c1_4_2_2_2_2_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7)) ) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_2_1__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7])],[e1_4_2_2_2_2_1__bhsp_7,e2_4_2_2_2_2_1__bhsp_7,i2_4_2_2_2_2_1__bhsp_7]), [interesting(0.05),file(bhsp_7,i1_4_2_2_2_2_1__bhsp_7),[file(bhsp_7,i1_4_2_2_2_2_1__bhsp_7)]]). fof(i1_4_2_2_2_2_1_tmp__bhsp_7,plain, ( m2_subset_1(c1_4_2_2_2_2_1__bhsp_7,k1_numbers,k5_numbers) => ( ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),c1_4_2_2_2_2_1__bhsp_7) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2_2_1__bhsp_7)) ) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2_2_1__bhsp_7,1))) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_2_2_2_1__bhsp_7])],[dt_c1_4_2_2_2_2_1__bhsp_7,i1_4_2_2_2_2_1__bhsp_7]), [interesting(0.2),e2_4_2_2_2_2__bhsp_7]). fof(e2_4_2_2_2_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),k1_nat_1(A,1)) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1))) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7])],[i1_4_2_2_2_2_1_tmp__bhsp_7,dh_c1_4_2_2_2_2_1__bhsp_7]), [interesting(0.2),file(bhsp_7,e2_4_2_2_2_2__bhsp_7),[file(bhsp_7,e2_4_2_2_2_2__bhsp_7)]]). fof(e3_4_2_2_2_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7])],[cc1_xreal_0,rc2_finset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_funct_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,commutativity_k2_xcmplx_0,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_4_2_2_2__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,rc1_funct_1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,s1_nat_1__e3_4_2_2_2_2__bhsp_7,e1_4_2_2_2_2__bhsp_7,e2_4_2_2_2_2__bhsp_7]), [interesting(0.2),file(bhsp_7,e3_4_2_2_2_2__bhsp_7),[file(bhsp_7,e3_4_2_2_2_2__bhsp_7)]]). fof(i1_4_2_2_2_2__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i1_4_2_2_2_2__bhsp_7)]), [interesting(0.2),trivial,file(bhsp_7,i1_4_2_2_2_2__bhsp_7)]). fof(e6_4_2_2_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7])],[e3_4_2_2_2_2__bhsp_7,i1_4_2_2_2_2__bhsp_7]), [interesting(0.35),file(bhsp_7,e6_4_2_2_2__bhsp_7),[file(bhsp_7,e6_4_2_2_2__bhsp_7)]]). fof(e2_4_2_2_2__bhsp_7,plain,( k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e5_4_2_2__bhsp_7]), [interesting(0.35),file(bhsp_7,e2_4_2_2_2__bhsp_7),[file(bhsp_7,e2_4_2_2_2__bhsp_7)]]). fof(e3_4_2_2_2__bhsp_7,plain, ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e2_4_2_2__bhsp_7]), [interesting(0.35),file(bhsp_7,e3_4_2_2_2__bhsp_7),[file(bhsp_7,e3_4_2_2_2__bhsp_7)]]). fof(d1_xboole_0,definition,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), [interesting(0.9),axiom,file(xboole_0,d1_xboole_0)]). fof(e4_4_2_2_2__bhsp_7,plain,( ? [A] : r2_hidden(A,k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e1_4_2_2_2__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc1_xboole_0,fc2_membered,fc3_binop_2,fc6_membered,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e1_4_2_2_2__bhsp_7,d1_xboole_0]), [interesting(0.35),file(bhsp_7,e4_4_2_2_2__bhsp_7),[file(bhsp_7,e4_4_2_2_2__bhsp_7)]]). fof(dh_c1_4_2_2_2_1__bhsp_7,definition, ( ( ( v1_finset_1(c1_4_2_2_2_1__bhsp_7) & m1_subset_1(c1_4_2_2_2_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(c1_4_2_2_2_1__bhsp_7) & r1_tarski(c1_4_2_2_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c1_4_2_2_2_1__bhsp_7) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_2_2_1__bhsp_7,c3_4__bhsp_7))) ) ) => ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) ), introduced(definition,[new_symbol(c1_4_2_2_2_1__bhsp_7),file(bhsp_7,c1_4_2_2_2_1__bhsp_7)]), [interesting(0.2),axiom,file(bhsp_7,c1_4_2_2_2_1__bhsp_7)]). fof(e1_4_2_2_2_1__bhsp_7,assumption, ( ~ v1_xboole_0(c1_4_2_2_2_1__bhsp_7) & r1_tarski(c1_4_2_2_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c1_4_2_2_2_1__bhsp_7) ), introduced(assumption,[file(bhsp_7,e1_4_2_2_2_1__bhsp_7)]), [interesting(0.2),axiom,file(bhsp_7,e1_4_2_2_2_1__bhsp_7)]). fof(dt_c1_4_2_2_2_1__bhsp_7,assumption, ( v1_finset_1(c1_4_2_2_2_1__bhsp_7) & m1_subset_1(c1_4_2_2_2_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), introduced(assumption,[file(bhsp_7,c1_4_2_2_2_1__bhsp_7)]), [interesting(0.2),axiom,file(bhsp_7,c1_4_2_2_2_1__bhsp_7)]). fof(e2_4_2_2_2_1__bhsp_7,plain,( k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)) = k2_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e5_4_2_2__bhsp_7]), [interesting(0.2),file(bhsp_7,e2_4_2_2_2_1__bhsp_7),[file(bhsp_7,e2_4_2_2_2_1__bhsp_7)]]). fof(e3_4_2_2_2_1__bhsp_7,plain,( r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc2_finset_1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc2_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,t3_subset,spc1_boole,spc1_numerals,e2_4_2_2_2_1__bhsp_7,t7_xboole_1]), [interesting(0.2),file(bhsp_7,e3_4_2_2_2_1__bhsp_7),[file(bhsp_7,e3_4_2_2_2_1__bhsp_7)]]). fof(t63_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_xboole_0(B,C) ) => r1_xboole_0(A,C) ) ), file(xboole_1,t63_xboole_1), [interesting(0.9),axiom,file(xboole_1,t63_xboole_1)]). fof(e4_4_2_2_2_1__bhsp_7,plain, ( ~ v1_xboole_0(c1_4_2_2_2_1__bhsp_7) & r1_tarski(c1_4_2_2_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c1_4_2_2_2_1__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_2_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l1_bhsp_1,existence_l1_struct_0,dt_k2_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,cc1_funct_2,cc1_relset_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc2_finset_1,rc3_funct_1,rc3_struct_0,rc5_struct_0,spc6_arithm,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc2_membered,fc5_membered,fc6_membered,rc1_finset_1,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t4_subset,t5_subset,t8_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k8_funct_2,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,rc1_xboole_0,rc2_xboole_0,spc1_boole,t3_subset,t6_boole,t7_boole,spc1_boole,spc1_numerals,e3_4_2_2_2_1__bhsp_7,e1_4_2_2_2_1__bhsp_7,t63_xboole_1]), [interesting(0.2),file(bhsp_7,e4_4_2_2_2_1__bhsp_7),[file(bhsp_7,e4_4_2_2_2_1__bhsp_7)]]). fof(e5_4_2_2_2_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_2_2_1__bhsp_7,c3_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,e1_4_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,spc6_arithm,t1_real,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2_1__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,spc1_boole,spc2_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e4_4_2_2_2_1__bhsp_7,e2_4_2_2__bhsp_7,rqRealAdd__k2_xcmplx_0__r1_r1_r2]), [interesting(0.2),file(bhsp_7,e5_4_2_2_2_1__bhsp_7),[file(bhsp_7,e5_4_2_2_2_1__bhsp_7)]]). fof(i3_4_2_2_2_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_2_2_1__bhsp_7)]), [interesting(0.2),trivial,file(bhsp_7,i3_4_2_2_2_1__bhsp_7)]). fof(i2_4_2_2_2_1__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_2_2_1__bhsp_7,c3_4__bhsp_7))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,e1_4_2_2_2_1__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[e5_4_2_2_2_1__bhsp_7,i3_4_2_2_2_1__bhsp_7]), [interesting(0.2),file(bhsp_7,i2_4_2_2_2_1__bhsp_7),[file(bhsp_7,i2_4_2_2_2_1__bhsp_7)]]). fof(i1_4_2_2_2_1__bhsp_7,plain,( ~ ( ~ v1_xboole_0(c1_4_2_2_2_1__bhsp_7) & r1_tarski(c1_4_2_2_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c1_4_2_2_2_1__bhsp_7) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_2_2_1__bhsp_7,c3_4__bhsp_7))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_2_1__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_2_1__bhsp_7])],[e1_4_2_2_2_1__bhsp_7,i2_4_2_2_2_1__bhsp_7]), [interesting(0.2),file(bhsp_7,i1_4_2_2_2_1__bhsp_7),[file(bhsp_7,i1_4_2_2_2_1__bhsp_7)]]). fof(i1_4_2_2_2_1_tmp__bhsp_7,plain, ( ( v1_finset_1(c1_4_2_2_2_1__bhsp_7) & m1_subset_1(c1_4_2_2_2_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(c1_4_2_2_2_1__bhsp_7) & r1_tarski(c1_4_2_2_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c1_4_2_2_2_1__bhsp_7) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_2_2_1__bhsp_7,c3_4__bhsp_7))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_2_2_1__bhsp_7])],[dt_c1_4_2_2_2_1__bhsp_7,i1_4_2_2_2_1__bhsp_7]), [interesting(0.35),e5_4_2_2_2__bhsp_7]). fof(e5_4_2_2_2__bhsp_7,plain,( ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[i1_4_2_2_2_1_tmp__bhsp_7,dh_c1_4_2_2_2_1__bhsp_7]), [interesting(0.35),file(bhsp_7,e5_4_2_2_2__bhsp_7),[file(bhsp_7,e5_4_2_2_2__bhsp_7)]]). fof(t14_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,t14_finset_1), [interesting(0.9),axiom,file(finset_1,t14_finset_1)]). fof(d2_xboole_0,definition,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), [interesting(0.9),axiom,file(xboole_0,d2_xboole_0)]). fof(e7_4_2_2_2__bhsp_7,plain, ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(mizar_by,[status(thm),assumptions([e1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc23_xreal_0,fc24_membered,fc25_membered,fc26_membered,fc30_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_boole,t1_real,t3_arithm,t4_real,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc2_xboole_0,fc3_binop_2,fc3_xboole_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,spc1_boole,spc2_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e6_4_2_2_2__bhsp_7,e1_4_2_2_2__bhsp_7,e2_4_2_2_2__bhsp_7,e3_4_2_2_2__bhsp_7,e4_4_2_2_2__bhsp_7,e5_4_2_2_2__bhsp_7,t14_finset_1,d2_xboole_0,t8_xboole_1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealMult__k3_xcmplx_0__r2_r1_r2]), [interesting(0.35),file(bhsp_7,e7_4_2_2_2__bhsp_7),[file(bhsp_7,e7_4_2_2_2__bhsp_7)]]). fof(i3_4_2_2_2__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_2_2__bhsp_7)]), [interesting(0.35),trivial,file(bhsp_7,i3_4_2_2_2__bhsp_7)]). fof(i2_4_2_2_2__bhsp_7,plain, ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ), inference(conclusion,[status(thm),assumptions([e1_4_2_2_2__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k2_xcmplx_0,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,spc1_numerals,e7_4_2_2_2__bhsp_7,i3_4_2_2_2__bhsp_7]), [interesting(0.35),file(bhsp_7,i2_4_2_2_2__bhsp_7),[file(bhsp_7,i2_4_2_2_2__bhsp_7)]]). fof(i2_4_2_2_2_tmp__bhsp_7,plain, ( ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_2__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),B) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),B) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,B)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_2__bhsp_7])],[e1_4_2_2_2__bhsp_7,i2_4_2_2_2__bhsp_7]), [interesting(0.35),i1_4_2_2_2__bhsp_7]). fof(i1_4_2_2_2__bhsp_7,plain, ( ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_2__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_4_2_2_2__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[i2_4_2_2_2_tmp__bhsp_7,dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k2_xcmplx_0,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,spc1_numerals]), [interesting(0.35),file(bhsp_7,i1_4_2_2_2__bhsp_7),[file(bhsp_7,i1_4_2_2_2__bhsp_7)]]). fof(i1_4_2_2_2_tmp__bhsp_7,plain, ( m2_subset_1(c1_4_2_2_2__bhsp_7,k1_numbers,k5_numbers) => ( ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c1_4_2_2_2__bhsp_7,A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c1_4_2_2_2__bhsp_7),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(c1_4_2_2_2__bhsp_7,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(c1_4_2_2_2__bhsp_7,1),A) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(c1_4_2_2_2__bhsp_7,1)),k1_funct_1(c2_4_2_2__bhsp_7,A)) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_2_2__bhsp_7])],[dt_c1_4_2_2_2__bhsp_7,i1_4_2_2_2__bhsp_7]), [interesting(0.5),e8_4_2_2__bhsp_7]). fof(e8_4_2_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,A)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,A),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,A)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,A),c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,A),B) & r1_xreal_0(k6_real_1(1,k1_nat_1(A,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,A),k1_funct_1(c2_4_2_2__bhsp_7,B)) ) ) ) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1))) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1)),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1))) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1)),c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1)),B) & r1_xreal_0(k6_real_1(1,k1_nat_1(k1_nat_1(A,1),1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(k1_nat_1(A,1),B) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,k1_nat_1(A,1)),k1_funct_1(c2_4_2_2__bhsp_7,B)) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[i1_4_2_2_2_tmp__bhsp_7,dh_c1_4_2_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e8_4_2_2__bhsp_7),[file(bhsp_7,e8_4_2_2__bhsp_7)]]). fof(e9_4_2_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( v1_finset_1(k1_funct_1(c2_4_2_2__bhsp_7,A)) & m1_subset_1(k1_funct_1(c2_4_2_2__bhsp_7,A),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,A)) & r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,A),c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c2_4_2_2__bhsp_7,A),B) & r1_xreal_0(k6_real_1(1,k1_nat_1(A,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,A),k1_funct_1(c2_4_2_2__bhsp_7,B)) ) ) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k2_xcmplx_0,dt_k16_complex1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_k7_xcmplx_0,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_2,rc1_membered,rc1_xreal_0,rc3_funct_1,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc2_funct_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,s1_nat_1__e9_4_2_2__bhsp_7,e7_4_2_2__bhsp_7,e8_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e9_4_2_2__bhsp_7),[file(bhsp_7,e9_4_2_2__bhsp_7)]]). fof(e5_4_2_2_3__bhsp_7,plain,( r1_tarski(k1_funct_1(c2_4_2_2__bhsp_7,c3_4_2_2_3__bhsp_7),c2_4__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,e1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_c2_4_2_2_3__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_2_3__bhsp_7,de_c3_4_2_2_3__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc1_boole,spc1_numerals,e9_4_2_2__bhsp_7]), [interesting(0.35),file(bhsp_7,e5_4_2_2_3__bhsp_7),[file(bhsp_7,e5_4_2_2_3__bhsp_7)]]). fof(e6_4_2_2_3__bhsp_7,plain,( r2_hidden(c1_4_2_2_3__bhsp_7,k1_zfmisc_1(c2_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc2_finset_1,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc6_membered,cc9_membered,fc2_membered,rc1_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_c1_4_2_2_3__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c2_4_2_2_3__bhsp_7,dt_c3_4_2_2_3__bhsp_7,de_c3_4_2_2_3__bhsp_7,t1_subset,t3_subset,t7_boole,e5_4_2_2_3__bhsp_7,e3_4_2_2_3__bhsp_7,d1_zfmisc_1]), [interesting(0.35),file(bhsp_7,e6_4_2_2_3__bhsp_7),[file(bhsp_7,e6_4_2_2_3__bhsp_7)]]). fof(i3_4_2_2_3__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_2_3__bhsp_7)]), [interesting(0.35),trivial,file(bhsp_7,i3_4_2_2_3__bhsp_7)]). fof(i2_4_2_2_3__bhsp_7,plain,( r2_hidden(c1_4_2_2_3__bhsp_7,k1_zfmisc_1(c2_4__bhsp_7)) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7,e1_4_2_2_3__bhsp_7])],[e6_4_2_2_3__bhsp_7,i3_4_2_2_3__bhsp_7]), [interesting(0.35),file(bhsp_7,i2_4_2_2_3__bhsp_7),[file(bhsp_7,i2_4_2_2_3__bhsp_7)]]). fof(i1_4_2_2_3__bhsp_7,plain,( ~ ( r2_hidden(c1_4_2_2_3__bhsp_7,k2_relat_1(c2_4_2_2__bhsp_7)) & ~ r2_hidden(c1_4_2_2_3__bhsp_7,k1_zfmisc_1(c2_4__bhsp_7)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_2_3__bhsp_7])],[e1_4_2_2_3__bhsp_7,i2_4_2_2_3__bhsp_7]), [interesting(0.35),file(bhsp_7,i1_4_2_2_3__bhsp_7),[file(bhsp_7,i1_4_2_2_3__bhsp_7)]]). fof(i1_4_2_2_3_tmp__bhsp_7,plain,( ~ ( r2_hidden(c1_4_2_2_3__bhsp_7,k2_relat_1(c2_4_2_2__bhsp_7)) & ~ r2_hidden(c1_4_2_2_3__bhsp_7,k1_zfmisc_1(c2_4__bhsp_7)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_2_3__bhsp_7])],[dt_c1_4_2_2_3__bhsp_7,i1_4_2_2_3__bhsp_7]), [interesting(0.5),e10_4_2_2__bhsp_7]). fof(e10_4_2_2__bhsp_7,plain,( r1_tarski(k2_relat_1(c2_4_2_2__bhsp_7),k1_zfmisc_1(c2_4__bhsp_7)) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[i1_4_2_2_3_tmp__bhsp_7,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,dt_l2_struct_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc4_struct_0,dt_l1_rlvect_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_funct_1,rc4_finset_1,dt_l2_rlvect_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_funct_1,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rc5_struct_0,dt_l1_bhsp_1,dt_l1_struct_0,fc1_struct_0,rc3_struct_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,rc1_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_zfmisc_1,dt_k2_relat_1,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,d3_tarski,dh_c1_4_2_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,e10_4_2_2__bhsp_7),[file(bhsp_7,e10_4_2_2__bhsp_7)]]). fof(t4_funct_2,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(k2_relat_1(B),A) => ( v1_funct_1(B) & v1_funct_2(B,k1_relat_1(B),A) & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ), file(funct_2,t4_funct_2), [interesting(0.9),axiom,file(funct_2,t4_funct_2)]). fof(e11_4_2_2__bhsp_7,plain, ( v1_funct_1(c2_4_2_2__bhsp_7) & v1_funct_2(c2_4_2_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(c2_4_2_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_l1_bhsp_1,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc24_membered,fc25_membered,fc26_membered,fc3_xreal_0,fc6_membered,fc8_xreal_0,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc6_arithm,t1_arithm,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_funct_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc1_ordinal2,fc22_membered,fc23_membered,fc2_xboole_0,fc3_xboole_0,fc5_membered,rc1_funct_2,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_relat_1,dt_k2_xboole_0,dt_k5_numbers,dt_k8_funct_2,dt_m2_relset_1,dt_m2_subset_1,dt_c1_4_2_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,fc2_membered,rc1_funct_1,t3_subset,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e10_4_2_2__bhsp_7,e5_4_2_2__bhsp_7,t4_funct_2]), [interesting(0.5),file(bhsp_7,e11_4_2_2__bhsp_7),[file(bhsp_7,e11_4_2_2__bhsp_7)]]). fof(e12_4_2_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),C) & r1_xreal_0(k6_real_1(1,k1_nat_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,C)) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc1_boole,spc1_numerals,e11_4_2_2__bhsp_7,e9_4_2_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e12_4_2_2__bhsp_7),[file(bhsp_7,e12_4_2_2__bhsp_7)]]). fof(i1_4_2_2__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i1_4_2_2__bhsp_7)]), [interesting(0.5),trivial,file(bhsp_7,i1_4_2_2__bhsp_7)]). fof(e2_4_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),C) & r1_xreal_0(k6_real_1(1,k1_nat_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,C)) ) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k2_xcmplx_0,redefinition_k6_supinf_1,dt_k16_complex1,dt_k1_funct_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,spc1_numerals,e12_4_2_2__bhsp_7,i1_4_2_2__bhsp_7]), [interesting(0.65),file(bhsp_7,e2_4_2__bhsp_7),[file(bhsp_7,e2_4_2__bhsp_7)]]). fof(e3_4_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(A,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),c2_4__bhsp_7) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & r1_tarski(C,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),C) & r1_xreal_0(k6_real_1(1,k1_nat_1(B,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7))) ) ) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(B,C) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,B),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),A,C)) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc1_boole,spc1_numerals,e2_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,e3_4_2__bhsp_7),[file(bhsp_7,e3_4_2__bhsp_7)]]). fof(dt_c1_4_2__bhsp_7,plain, ( v1_funct_1(c1_4_2__bhsp_7) & v1_funct_2(c1_4_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) & m2_relset_1(c1_4_2__bhsp_7,k5_numbers,k1_zfmisc_1(c2_4__bhsp_7)) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c1_4_2__bhsp_7,e3_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,c1_4_2__bhsp_7),[file(bhsp_7,c1_4_2__bhsp_7)]]). fof(s1_funct_2__e6_4_2__bhsp_7,theorem,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v3_rlvect_1(A) & v4_rlvect_1(A) & v5_rlvect_1(A) & v6_rlvect_1(A) & v7_rlvect_1(A) & v2_bhsp_1(A) & l1_bhsp_1(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k6_supinf_1) & m2_relset_1(C,u1_struct_0(A),k6_supinf_1) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_zfmisc_1(B)) & m2_relset_1(D,k5_numbers,k1_zfmisc_1(B)) ) => ( ! [E] : ~ ( r2_hidden(E,k5_numbers) & ! [F] : ~ ( r2_hidden(F,k1_numbers) & ? [G] : ( v1_finset_1(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(G) & k1_funct_1(D,E) = G & F = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,G,C) ) ) ) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k5_numbers,k1_numbers) & m2_relset_1(E,k5_numbers,k1_numbers) & ! [F] : ( r2_hidden(F,k5_numbers) => ? [H] : ( v1_finset_1(H) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(H) & k1_funct_1(D,F) = H & k1_funct_1(E,F) = k5_bhsp_5(k1_numbers,u1_struct_0(A),k33_binop_2,H,C) ) ) ) ) ) ), file(bhsp_7,s1_funct_2__e6_4_2__bhsp_7), [interesting(0.9),axiom,file(bhsp_7,s1_funct_2__e6_4_2__bhsp_7)]). fof(dh_c1_4_2_3__bhsp_7,definition, ( ~ ( r2_hidden(c1_4_2_3__bhsp_7,k5_numbers) & ! [A] : ~ ( r2_hidden(A,k1_numbers) & ? [B] : ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) = B & A = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7) ) ) ) => ! [C] : ~ ( r2_hidden(C,k5_numbers) & ! [D] : ~ ( r2_hidden(D,k1_numbers) & ? [E] : ( v1_finset_1(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(E) & k1_funct_1(c1_4_2__bhsp_7,C) = E & D = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,E,c3_4__bhsp_7) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_3__bhsp_7),file(bhsp_7,c1_4_2_3__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_3__bhsp_7)]). fof(e1_4_2_3__bhsp_7,assumption,( r2_hidden(c1_4_2_3__bhsp_7,k5_numbers) ), introduced(assumption,[file(bhsp_7,e1_4_2_3__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,e1_4_2_3__bhsp_7)]). fof(dt_c1_4_2_3__bhsp_7,assumption,( $true ), introduced(assumption,[file(bhsp_7,c1_4_2_3__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_3__bhsp_7)]). fof(de_c3_4_2_3__bhsp_7,definition,( c3_4_2_3__bhsp_7 = k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) ), introduced(definition,[new_symbol(c3_4_2_3__bhsp_7),file(bhsp_7,c3_4_2_3__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c3_4_2_3__bhsp_7)]). fof(de_c2_4_2_3__bhsp_7,definition,( c2_4_2_3__bhsp_7 = c1_4_2_3__bhsp_7 ), introduced(definition,[new_symbol(c2_4_2_3__bhsp_7),file(bhsp_7,c2_4_2_3__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c2_4_2_3__bhsp_7)]). fof(e2_4_2_3__bhsp_7,plain,( m2_subset_1(c1_4_2_3__bhsp_7,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7])],[cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_4_2_3__bhsp_7,fc2_membered,t1_subset,t7_boole,e1_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,e2_4_2_3__bhsp_7),[file(bhsp_7,e2_4_2_3__bhsp_7)]]). fof(dt_c2_4_2_3__bhsp_7,plain,( m2_subset_1(c2_4_2_3__bhsp_7,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7])],[cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_4_2_3__bhsp_7,fc2_membered,de_c2_4_2_3__bhsp_7,e2_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,c2_4_2_3__bhsp_7),[file(bhsp_7,c2_4_2_3__bhsp_7)]]). fof(e4_4_2__bhsp_7,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A),c2_4__bhsp_7) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & r1_tarski(B,c2_4__bhsp_7) & r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A),B) & r1_xreal_0(k6_real_1(1,k1_nat_1(A,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7))) ) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( r1_xreal_0(A,B) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,B)) ) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,projectivity_k16_complex1,commutativity_k2_xcmplx_0,redefinition_k6_supinf_1,dt_k16_complex1,dt_k1_funct_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,spc1_numerals,dh_c1_4_2__bhsp_7,e3_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,e4_4_2__bhsp_7),[file(bhsp_7,e4_4_2__bhsp_7)]]). fof(e3_4_2_3__bhsp_7,plain, ( v1_finset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_3__bhsp_7)) & m1_subset_1(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_3__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_3__bhsp_7)) & r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_3__bhsp_7),c2_4__bhsp_7) & ! [A] : ( ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7),A) & r1_xreal_0(k6_real_1(1,k1_nat_1(c2_4_2_3__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7))) ) ) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(c2_4_2_3__bhsp_7,A) => r1_tarski(k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_3__bhsp_7),k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,A)) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_3__bhsp_7,dt_c3_4__bhsp_7,de_c2_4_2_3__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc1_boole,spc1_numerals,e4_4_2__bhsp_7,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(bhsp_7,e3_4_2_3__bhsp_7),[file(bhsp_7,e3_4_2_3__bhsp_7)]]). fof(e4_4_2_3__bhsp_7,plain, ( v1_finset_1(k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7)) & m1_subset_1(k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7),k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_3__bhsp_7,dt_c3_4__bhsp_7,de_c2_4_2_3__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,spc1_boole,spc1_numerals,e3_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,e4_4_2_3__bhsp_7),[file(bhsp_7,e4_4_2_3__bhsp_7)]]). fof(dt_c3_4_2_3__bhsp_7,plain, ( v1_finset_1(c3_4_2_3__bhsp_7) & m1_subset_1(c3_4_2_3__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,fc1_xboole_0,fc6_membered,rc1_xreal_0,rc2_finset_1,rc3_funct_1,t1_subset,t4_subset,t5_subset,existence_l2_rlvect_1,existence_m1_relset_1,dt_k1_numbers,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,fc14_finset_1,fc1_ordinal2,fc2_membered,fc5_membered,rc1_finset_1,rc1_funct_2,rc1_membered,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k5_numbers,dt_l1_bhsp_1,dt_l1_struct_0,dt_m2_relset_1,dt_c2_4__bhsp_7,cc9_membered,fc1_struct_0,rc1_funct_1,rc3_struct_0,existence_m1_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,cc2_finset_1,t3_subset,de_c3_4_2_3__bhsp_7,e4_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,c3_4_2_3__bhsp_7),[file(bhsp_7,c3_4_2_3__bhsp_7)]]). fof(de_c4_4_2_3__bhsp_7,definition,( c4_4_2_3__bhsp_7 = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c3_4_2_3__bhsp_7,c3_4__bhsp_7) ), introduced(definition,[new_symbol(c4_4_2_3__bhsp_7),file(bhsp_7,c4_4_2_3__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c4_4_2_3__bhsp_7)]). fof(e5_4_2_3__bhsp_7,plain,( $true ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc2_finset_1,rc4_struct_0,existence_l1_rlvect_1,dt_k5_ordinal2,dt_l1_rlvect_1,cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_ordinal2,fc5_membered,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,redefinition_k5_numbers,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_numbers,dt_l2_rlvect_1,dt_c2_4__bhsp_7,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,cc9_membered,fc1_xboole_0,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,t1_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,fc14_finset_1,fc1_struct_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,dt_k1_numbers,dt_k33_binop_2,dt_k5_bhsp_5,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_3__bhsp_7,de_c3_4_2_3__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2]), [interesting(0.5),file(bhsp_7,e5_4_2_3__bhsp_7),[file(bhsp_7,e5_4_2_3__bhsp_7)]]). fof(dt_c4_4_2_3__bhsp_7,plain,( $true ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc2_finset_1,rc4_struct_0,existence_l1_rlvect_1,dt_k5_ordinal2,dt_l1_rlvect_1,cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_ordinal2,fc5_membered,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,redefinition_k5_numbers,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_numbers,dt_l2_rlvect_1,dt_c2_4__bhsp_7,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,cc9_membered,fc1_xboole_0,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,t1_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,fc14_finset_1,fc1_struct_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,dt_k1_numbers,dt_k33_binop_2,dt_k5_bhsp_5,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_3__bhsp_7,de_c3_4_2_3__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,de_c4_4_2_3__bhsp_7,e5_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,c4_4_2_3__bhsp_7),[file(bhsp_7,c4_4_2_3__bhsp_7)]]). fof(e6_4_2_3__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) = A & c4_4_2_3__bhsp_7 = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_c3_4_2_3__bhsp_7,de_c3_4_2_3__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2_3__bhsp_7,dt_c3_4__bhsp_7,dt_c4_4_2_3__bhsp_7,de_c2_4_2_3__bhsp_7,de_c4_4_2_3__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e3_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,e6_4_2_3__bhsp_7),[file(bhsp_7,e6_4_2_3__bhsp_7)]]). fof(e7_4_2_3__bhsp_7,plain,( ? [A] : ( r2_hidden(A,k1_numbers) & ? [B] : ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) = B & A = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,fc1_ordinal2,fc5_membered,rc1_xreal_0,rc3_funct_1,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_c2_4__bhsp_7,dt_c3_4_2_3__bhsp_7,de_c3_4_2_3__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,cc9_membered,fc14_finset_1,fc1_struct_0,fc1_xboole_0,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_3__bhsp_7,dt_c3_4__bhsp_7,dt_c4_4_2_3__bhsp_7,de_c4_4_2_3__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,e6_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,e7_4_2_3__bhsp_7),[file(bhsp_7,e7_4_2_3__bhsp_7)]]). fof(i3_4_2_3__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i3_4_2_3__bhsp_7)]), [interesting(0.5),trivial,file(bhsp_7,i3_4_2_3__bhsp_7)]). fof(i2_4_2_3__bhsp_7,plain,( ? [A] : ( r2_hidden(A,k1_numbers) & ? [B] : ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) = B & A = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,e1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[e7_4_2_3__bhsp_7,i3_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,i2_4_2_3__bhsp_7),[file(bhsp_7,i2_4_2_3__bhsp_7)]]). fof(i1_4_2_3__bhsp_7,plain,( ~ ( r2_hidden(c1_4_2_3__bhsp_7,k5_numbers) & ! [A] : ~ ( r2_hidden(A,k1_numbers) & ? [B] : ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) = B & A = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_3__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_3__bhsp_7])],[e1_4_2_3__bhsp_7,i2_4_2_3__bhsp_7]), [interesting(0.5),file(bhsp_7,i1_4_2_3__bhsp_7),[file(bhsp_7,i1_4_2_3__bhsp_7)]]). fof(i1_4_2_3_tmp__bhsp_7,plain,( ~ ( r2_hidden(c1_4_2_3__bhsp_7,k5_numbers) & ! [A] : ~ ( r2_hidden(A,k1_numbers) & ? [B] : ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(B) & k1_funct_1(c1_4_2__bhsp_7,c1_4_2_3__bhsp_7) = B & A = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7]),discharge_asm(discharge,[dt_c1_4_2_3__bhsp_7])],[dt_c1_4_2_3__bhsp_7,i1_4_2_3__bhsp_7]), [interesting(0.65),e5_4_2__bhsp_7]). fof(e5_4_2__bhsp_7,plain,( ! [A] : ~ ( r2_hidden(A,k5_numbers) & ! [B] : ~ ( r2_hidden(B,k1_numbers) & ? [C] : ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(C) & k1_funct_1(c1_4_2__bhsp_7,A) = C & B = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[i1_4_2_3_tmp__bhsp_7,dh_c1_4_2_3__bhsp_7]), [interesting(0.65),file(bhsp_7,e5_4_2__bhsp_7),[file(bhsp_7,e5_4_2__bhsp_7)]]). fof(e6_4_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ( r2_hidden(B,k5_numbers) => ? [C] : ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(C) & k1_funct_1(c1_4_2__bhsp_7,B) = C & k1_funct_1(A,B) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7) ) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,dt_l1_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc2_finset_1,rc3_funct_1,dt_k2_zfmisc_1,dt_k3_supinf_1,dt_k5_ordinal2,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_c2_4__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc5_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,antisymmetry_r2_hidden,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,s1_funct_2__e6_4_2__bhsp_7,e5_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,e6_4_2__bhsp_7),[file(bhsp_7,e6_4_2__bhsp_7)]]). fof(e7_4_2__bhsp_7,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ~ ( r2_hidden(B,k5_numbers) & ! [C] : ( ( v1_finset_1(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(C) & k1_funct_1(c1_4_2__bhsp_7,B) = C & k1_funct_1(A,B) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,C,c3_4__bhsp_7) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_c2_4__bhsp_7,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,e6_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,e7_4_2__bhsp_7),[file(bhsp_7,e7_4_2__bhsp_7)]]). fof(dt_c2_4_2__bhsp_7,plain, ( v1_funct_1(c2_4_2__bhsp_7) & v1_funct_2(c2_4_2__bhsp_7,k5_numbers,k1_numbers) & m2_relset_1(c2_4_2__bhsp_7,k5_numbers,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c2_4_2__bhsp_7,e7_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,c2_4_2__bhsp_7),[file(bhsp_7,c2_4_2__bhsp_7)]]). fof(dh_c1_4_2_5__bhsp_7,definition, ( ( v1_xreal_0(c1_4_2_5__bhsp_7) => ~ ( ~ r1_xreal_0(c1_4_2_5__bhsp_7,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & r1_xreal_0(A,B) & r1_xreal_0(c1_4_2_5__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,B),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,A)))) ) ) ) ) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ? [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) & r1_xreal_0(D,E) & r1_xreal_0(C,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,E),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,D)))) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_5__bhsp_7),file(bhsp_7,c1_4_2_5__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_5__bhsp_7)]). fof(e1_4_2_5__bhsp_7,assumption,( ~ r1_xreal_0(c1_4_2_5__bhsp_7,0) ), introduced(assumption,[file(bhsp_7,e1_4_2_5__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,e1_4_2_5__bhsp_7)]). fof(dt_c1_4_2_5__bhsp_7,assumption,( v1_xreal_0(c1_4_2_5__bhsp_7) ), introduced(assumption,[file(bhsp_7,c1_4_2_5__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_5__bhsp_7)]). fof(dh_c2_4_2_5__bhsp_7,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c1_4_2_5__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,B),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,C)))) ) ) ) ) => ( m2_subset_1(c2_4_2_5__bhsp_7,k1_numbers,k5_numbers) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_4_2_5__bhsp_7,D) & r1_xreal_0(c2_4_2_5__bhsp_7,E) & r1_xreal_0(c1_4_2_5__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,D),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,E)))) ) ) ) ) ), introduced(definition,[new_symbol(c2_4_2_5__bhsp_7),file(bhsp_7,c2_4_2_5__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c2_4_2_5__bhsp_7)]). fof(dh_c1_4_2_4__bhsp_7,definition, ( ( v1_xreal_0(c1_4_2_4__bhsp_7) => ~ ( ~ r1_xreal_0(c1_4_2_4__bhsp_7,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(A,B) & r1_xreal_0(A,C) & r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,B),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,C)))) ) ) ) ) ) => ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & ? [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) & r1_xreal_0(E,F) & r1_xreal_0(E,G) & r1_xreal_0(D,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,F),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,G)))) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_4_2_4__bhsp_7),file(bhsp_7,c1_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_4__bhsp_7)]). fof(e1_4_2_4__bhsp_7,assumption,( ~ r1_xreal_0(c1_4_2_4__bhsp_7,0) ), introduced(assumption,[file(bhsp_7,e1_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,e1_4_2_4__bhsp_7)]). fof(dt_c1_4_2_4__bhsp_7,assumption,( v1_xreal_0(c1_4_2_4__bhsp_7) ), introduced(assumption,[file(bhsp_7,c1_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c1_4_2_4__bhsp_7)]). fof(dh_c2_4_2_4__bhsp_7,definition, ( ~ ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r1_xreal_0(k7_xcmplx_0(c1_4_2_4__bhsp_7,2),k6_real_1(1,k1_nat_1(A,1))) ) => ( m2_subset_1(c2_4_2_4__bhsp_7,k1_numbers,k5_numbers) & ~ r1_xreal_0(k7_xcmplx_0(c1_4_2_4__bhsp_7,2),k6_real_1(1,k1_nat_1(c2_4_2_4__bhsp_7,1))) ) ), introduced(definition,[new_symbol(c2_4_2_4__bhsp_7),file(bhsp_7,c2_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c2_4_2_4__bhsp_7)]). fof(rqRealAdd__k2_xcmplx_0__r0_rn1d2_rn1d2,theorem,( k2_xcmplx_0(0,k7_xcmplx_0(1,2)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rn1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rn1d2_rn1d2)]). fof(rqRealAdd__k2_xcmplx_0__rn1d2_r0_rn1d2,theorem,( k2_xcmplx_0(k7_xcmplx_0(1,2),0) = k7_xcmplx_0(1,2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_r0_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_r0_rn1d2)]). fof(e2_4_2_4__bhsp_7,plain,( ~ r1_xreal_0(k7_xcmplx_0(c1_4_2_4__bhsp_7,2),k6_real_1(0,2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_membered,fc30_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc4_arithm,spc7_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k6_real_1,dt_k3_xcmplx_0,dt_k6_real_1,dt_k7_xcmplx_0,dt_c1_4_2_4__bhsp_7,cc2_xreal_0,fc4_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rn1d2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r0,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e1_4_2_4__bhsp_7,t73_real_1,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2]), [interesting(0.5),file(bhsp_7,e2_4_2_4__bhsp_7),[file(bhsp_7,e2_4_2_4__bhsp_7)]]). fof(l2_bhsp_7,plain,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(A,k6_real_1(1,k1_nat_1(B,1))) ) ) ) ), file(bhsp_7,l2_bhsp_7), [interesting(0.9),axiom,file(bhsp_7,l2_bhsp_7)]). fof(e3_4_2_4__bhsp_7,plain,( ~ ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r1_xreal_0(k7_xcmplx_0(c1_4_2_4__bhsp_7,2),k6_real_1(1,k1_nat_1(A,1))) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7])],[cc1_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc23_xreal_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rn1d2_rn1d2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__rn1d2_r0_rn1d2,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k3_xcmplx_0,dt_k5_numbers,dt_k6_real_1,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_4_2_4__bhsp_7,cc2_xreal_0,fc2_membered,fc4_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rn1d2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r0,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e2_4_2_4__bhsp_7,l2_bhsp_7,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r0]), [interesting(0.5),file(bhsp_7,e3_4_2_4__bhsp_7),[file(bhsp_7,e3_4_2_4__bhsp_7)]]). fof(dt_c2_4_2_4__bhsp_7,plain,( m2_subset_1(c2_4_2_4__bhsp_7,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7])],[dh_c2_4_2_4__bhsp_7,e3_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,c2_4_2_4__bhsp_7),[file(bhsp_7,c2_4_2_4__bhsp_7)]]). fof(dh_c3_4_2_4__bhsp_7,definition, ( ( m2_subset_1(c3_4_2_4__bhsp_7,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_4_2_4__bhsp_7,c3_4_2_4__bhsp_7) & r1_xreal_0(c2_4_2_4__bhsp_7,A) & r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,A)))) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_4_2_4__bhsp_7,B) & r1_xreal_0(c2_4_2_4__bhsp_7,C) & r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,B),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,C)))) ) ) ) ), introduced(definition,[new_symbol(c3_4_2_4__bhsp_7),file(bhsp_7,c3_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c3_4_2_4__bhsp_7)]). fof(dh_c4_4_2_4__bhsp_7,definition, ( ( m2_subset_1(c4_4_2_4__bhsp_7,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_4_2_4__bhsp_7,c3_4_2_4__bhsp_7) & r1_xreal_0(c2_4_2_4__bhsp_7,c4_4_2_4__bhsp_7) & r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_4_2_4__bhsp_7,c3_4_2_4__bhsp_7) & r1_xreal_0(c2_4_2_4__bhsp_7,A) & r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,A)))) ) ) ), introduced(definition,[new_symbol(c4_4_2_4__bhsp_7),file(bhsp_7,c4_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c4_4_2_4__bhsp_7)]). fof(e5_4_2_4__bhsp_7,assumption, ( r1_xreal_0(c2_4_2_4__bhsp_7,c3_4_2_4__bhsp_7) & r1_xreal_0(c2_4_2_4__bhsp_7,c4_4_2_4__bhsp_7) ), introduced(assumption,[file(bhsp_7,e5_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,e5_4_2_4__bhsp_7)]). fof(dt_c3_4_2_4__bhsp_7,assumption,( m2_subset_1(c3_4_2_4__bhsp_7,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_7,c3_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c3_4_2_4__bhsp_7)]). fof(dt_c4_4_2_4__bhsp_7,assumption,( m2_subset_1(c4_4_2_4__bhsp_7,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_7,c4_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c4_4_2_4__bhsp_7)]). fof(dh_c5_4_2_4__bhsp_7,definition, ( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_4__bhsp_7) = A & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c2_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) => ( v1_finset_1(c5_4_2_4__bhsp_7) & m1_subset_1(c5_4_2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(c5_4_2_4__bhsp_7) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_4__bhsp_7) = c5_4_2_4__bhsp_7 & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c2_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c5_4_2_4__bhsp_7,c3_4__bhsp_7) ) ), introduced(definition,[new_symbol(c5_4_2_4__bhsp_7),file(bhsp_7,c5_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c5_4_2_4__bhsp_7)]). fof(e8_4_2__bhsp_7,plain,( ! [A] : ~ ( r2_hidden(A,k5_numbers) & ! [B] : ( ( v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ) => ~ ( ~ v1_xboole_0(B) & k1_funct_1(c1_4_2__bhsp_7,A) = B & k1_funct_1(c2_4_2__bhsp_7,A) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,B,c3_4__bhsp_7) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c2_4_2__bhsp_7,e7_4_2__bhsp_7]), [interesting(0.65),file(bhsp_7,e8_4_2__bhsp_7),[file(bhsp_7,e8_4_2__bhsp_7)]]). fof(e6_4_2_4__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_4__bhsp_7) = A & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c2_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k8_funct_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c2_4_2_4__bhsp_7,dt_c3_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,e8_4_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e6_4_2_4__bhsp_7),[file(bhsp_7,e6_4_2_4__bhsp_7)]]). fof(dt_c5_4_2_4__bhsp_7,plain, ( v1_finset_1(c5_4_2_4__bhsp_7) & m1_subset_1(c5_4_2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(consider,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c5_4_2_4__bhsp_7,e6_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,c5_4_2_4__bhsp_7),[file(bhsp_7,c5_4_2_4__bhsp_7)]]). fof(dh_c6_4_2_4__bhsp_7,definition, ( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c3_4_2_4__bhsp_7) = A & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) => ( v1_finset_1(c6_4_2_4__bhsp_7) & m1_subset_1(c6_4_2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(c6_4_2_4__bhsp_7) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c3_4_2_4__bhsp_7) = c6_4_2_4__bhsp_7 & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c6_4_2_4__bhsp_7,c3_4__bhsp_7) ) ), introduced(definition,[new_symbol(c6_4_2_4__bhsp_7),file(bhsp_7,c6_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c6_4_2_4__bhsp_7)]). fof(e8_4_2_4__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c3_4_2_4__bhsp_7) = A & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k8_funct_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,e8_4_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e8_4_2_4__bhsp_7),[file(bhsp_7,e8_4_2_4__bhsp_7)]]). fof(dt_c6_4_2_4__bhsp_7,plain, ( v1_finset_1(c6_4_2_4__bhsp_7) & m1_subset_1(c6_4_2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(consider,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c6_4_2_4__bhsp_7,e8_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,c6_4_2_4__bhsp_7),[file(bhsp_7,c6_4_2_4__bhsp_7)]]). fof(e1_4_2_4_1_1_1__bhsp_7,assumption,( c5_4_2_4__bhsp_7 = c6_4_2_4__bhsp_7 ), introduced(assumption,[file(bhsp_7,e1_4_2_4_1_1_1__bhsp_7)]), [interesting(0.05),axiom,file(bhsp_7,e1_4_2_4_1_1_1__bhsp_7)]). fof(dh_c7_4_2_4__bhsp_7,definition, ( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c4_4_2_4__bhsp_7) = A & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) => ( v1_finset_1(c7_4_2_4__bhsp_7) & m1_subset_1(c7_4_2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(c7_4_2_4__bhsp_7) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c4_4_2_4__bhsp_7) = c7_4_2_4__bhsp_7 & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c7_4_2_4__bhsp_7,c3_4__bhsp_7) ) ), introduced(definition,[new_symbol(c7_4_2_4__bhsp_7),file(bhsp_7,c7_4_2_4__bhsp_7)]), [interesting(0.5),axiom,file(bhsp_7,c7_4_2_4__bhsp_7)]). fof(e10_4_2_4__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c4_4_2_4__bhsp_7) = A & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,A,c3_4__bhsp_7) ) ), inference(mizar_by,[status(thm),assumptions([dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_numbers,redefinition_k8_funct_2,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k8_funct_2,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c3_4__bhsp_7,dt_c4_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,e8_4_2__bhsp_7]), [interesting(0.5),file(bhsp_7,e10_4_2_4__bhsp_7),[file(bhsp_7,e10_4_2_4__bhsp_7)]]). fof(dt_c7_4_2_4__bhsp_7,plain, ( v1_finset_1(c7_4_2_4__bhsp_7) & m1_subset_1(c7_4_2_4__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(consider,[status(thm),assumptions([dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c7_4_2_4__bhsp_7,e10_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,c7_4_2_4__bhsp_7),[file(bhsp_7,c7_4_2_4__bhsp_7)]]). fof(e1_4_2_4_1_1_1_1_1_1__bhsp_7,assumption,( c5_4_2_4__bhsp_7 = c7_4_2_4__bhsp_7 ), introduced(assumption,[file(bhsp_7,e1_4_2_4_1_1_1_1_1_1__bhsp_7)]), [interesting(0.02),axiom,file(bhsp_7,e1_4_2_4_1_1_1_1_1_1__bhsp_7)]). fof(e9_4_2_4__bhsp_7,plain, ( ~ v1_xboole_0(c6_4_2_4__bhsp_7) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c3_4_2_4__bhsp_7) = c6_4_2_4__bhsp_7 & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c6_4_2_4__bhsp_7,c3_4__bhsp_7) ), inference(consider,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c6_4_2_4__bhsp_7,e8_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,e9_4_2_4__bhsp_7),[file(bhsp_7,e9_4_2_4__bhsp_7)]]). fof(e11_4_2_4__bhsp_7,plain, ( ~ v1_xboole_0(c7_4_2_4__bhsp_7) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c4_4_2_4__bhsp_7) = c7_4_2_4__bhsp_7 & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c7_4_2_4__bhsp_7,c3_4__bhsp_7) ), inference(consider,[status(thm),assumptions([dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c7_4_2_4__bhsp_7,e10_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,e11_4_2_4__bhsp_7),[file(bhsp_7,e11_4_2_4__bhsp_7)]]). fof(t7_absvalue,theorem,( ! [A] : ( v1_xreal_0(A) => ( A = 0 <=> k18_complex1(A) = 0 ) ) ), file(absvalue,t7_absvalue), [interesting(0.9),axiom,file(absvalue,t7_absvalue)]). fof(e2_4_2_4_1_1_1_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_1__bhsp_7,e1_4_2_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,e1_4_2_4_1_1_1__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,rc1_funct_1,rc1_funct_2,rc2_funct_1,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_funct_2,cc5_xreal_0,cc6_funct_2,cc6_membered,cc6_xreal_0,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_finset_1,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc5_membered,fc5_xreal_0,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t1_subset,t2_real,t2_subset,t3_real,t3_subset,t4_arithm,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k18_complex1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k8_funct_2,dt_k18_complex1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k33_binop_2,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_numbers,dt_k5_real_1,dt_k8_funct_2,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_4__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c6_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc7_xreal_0,fc12_binop_2,fc1_xreal_0,fc2_membered,fc3_binop_2,fc3_xreal_0,rc1_xboole_0,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_boole,spc1_boole,t6_boole,t7_boole,t8_boole,spc0_boole,spc1_boole,spc0_numerals,spc1_numerals,e1_4_2_4_1_1_1_1_1_1__bhsp_7,e1_4_2_4__bhsp_7,e9_4_2_4__bhsp_7,e11_4_2_4__bhsp_7,e1_4_2_4_1_1_1__bhsp_7,t7_absvalue,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0]), [interesting(0.02),file(bhsp_7,e2_4_2_4_1_1_1_1_1_1__bhsp_7),[file(bhsp_7,e2_4_2_4_1_1_1_1_1_1__bhsp_7)]]). fof(i2_4_2_4_1_1_1_1_1_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i2_4_2_4_1_1_1_1_1_1__bhsp_7)]), [interesting(0.02),trivial,file(bhsp_7,i2_4_2_4_1_1_1_1_1_1__bhsp_7)]). fof(i1_4_2_4_1_1_1_1_1_1__bhsp_7,plain,( ~ r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ), inference(conclusion,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_1__bhsp_7,e1_4_2_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,e1_4_2_4_1_1_1__bhsp_7])],[e2_4_2_4_1_1_1_1_1_1__bhsp_7,i2_4_2_4_1_1_1_1_1_1__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_4_1_1_1_1_1_1__bhsp_7),[file(bhsp_7,i1_4_2_4_1_1_1_1_1_1__bhsp_7)]]). fof(i1_4_2_4_1_1_1_1_1__bhsp_7,plain, ( c5_4_2_4__bhsp_7 = c7_4_2_4__bhsp_7 => ( c5_4_2_4__bhsp_7 = c7_4_2_4__bhsp_7 & ~ r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,e1_4_2_4_1_1_1__bhsp_7]),discharge_asm(discharge,[e1_4_2_4_1_1_1_1_1_1__bhsp_7])],[e1_4_2_4_1_1_1_1_1_1__bhsp_7,i1_4_2_4_1_1_1_1_1_1__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_4_1_1_1_1_1__bhsp_7),[file(bhsp_7,i1_4_2_4_1_1_1_1_1__bhsp_7)]]). fof(e1_4_2_4_1_1_1_1_1_2__bhsp_7,assumption,( c5_4_2_4__bhsp_7 != c7_4_2_4__bhsp_7 ), introduced(assumption,[file(bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7)]), [interesting(0.02),axiom,file(bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7)]). fof(dh_c1_4_2_4_1_1_1_1_1_2__bhsp_7,definition, ( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(A,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,A) = c7_4_2_4__bhsp_7 ) => ( v1_finset_1(c1_4_2_4_1_1_1_1_1_2__bhsp_7) & m1_subset_1(c1_4_2_4_1_1_1_1_1_2__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(c1_4_2_4_1_1_1_1_1_2__bhsp_7) & r1_tarski(c1_4_2_4_1_1_1_1_1_2__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c1_4_2_4_1_1_1_1_1_2__bhsp_7,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c1_4_2_4_1_1_1_1_1_2__bhsp_7) = c7_4_2_4__bhsp_7 ) ), introduced(definition,[new_symbol(c1_4_2_4_1_1_1_1_1_2__bhsp_7),file(bhsp_7,c1_4_2_4_1_1_1_1_1_2__bhsp_7)]), [interesting(0.02),axiom,file(bhsp_7,c1_4_2_4_1_1_1_1_1_2__bhsp_7)]). fof(fc37_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k4_xboole_0(A,B)) ) ), file(membered,fc37_membered), [interesting(0.9),axiom,file(membered,fc37_membered)]). fof(fc38_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc38_membered), [interesting(0.9),axiom,file(membered,fc38_membered)]). fof(fc39_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc39_membered), [interesting(0.9),axiom,file(membered,fc39_membered)]). fof(fc40_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc40_membered), [interesting(0.9),axiom,file(membered,fc40_membered)]). fof(fc41_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) & v5_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc41_membered), [interesting(0.9),axiom,file(membered,fc41_membered)]). fof(t3_boole,theorem,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), [interesting(0.9),axiom,file(boole,t3_boole)]). fof(t4_boole,theorem,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), [interesting(0.9),axiom,file(boole,t4_boole)]). fof(dt_k4_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), [interesting(0.9),axiom,file(xboole_0,k4_xboole_0)]). fof(fc12_finset_1,theorem,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), [interesting(0.9),axiom,file(finset_1,fc12_finset_1)]). fof(redefinition_k6_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) ), file(subset_1,k6_subset_1), [interesting(0.9),axiom,file(subset_1,k6_subset_1)]). fof(dt_k6_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k6_subset_1), [interesting(0.9),axiom,file(subset_1,k6_subset_1)]). fof(de_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,definition,( c1_4_2_4_1_1_1_1_1_2_1__bhsp_7 = k6_subset_1(u1_struct_0(c1_4__bhsp_7),c7_4_2_4__bhsp_7,c5_4_2_4__bhsp_7) ), introduced(definition,[new_symbol(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7),file(bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7)]), [interesting(0.02),axiom,file(bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7)]). fof(dt_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,plain,( m1_subset_1(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_struct_0,dt_l2_struct_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc4_struct_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_xboole_0,fc37_membered,fc38_membered,fc39_membered,fc40_membered,fc41_membered,fc6_membered,rc1_membered,t1_subset,t3_boole,t4_boole,t4_subset,t5_subset,existence_l2_rlvect_1,dt_l2_rlvect_1,cc15_membered,cc1_finset_1,cc1_funct_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,existence_l1_bhsp_1,existence_l1_struct_0,dt_k4_xboole_0,dt_l1_bhsp_1,dt_l1_struct_0,cc2_finset_1,fc12_finset_1,fc1_struct_0,rc3_struct_0,existence_m1_subset_1,redefinition_k6_subset_1,dt_k1_zfmisc_1,dt_k6_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,t3_subset,de_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7]), [interesting(0.02),file(bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7),[file(bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7)]]). fof(t39_xboole_1,theorem,( ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ), file(xboole_1,t39_xboole_1), [interesting(0.9),axiom,file(xboole_1,t39_xboole_1)]). fof(e1_4_2_4_1_1_1_1_1_2_1_1__bhsp_7,plain,( k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7) = k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c7_4_2_4__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_struct_0,dt_l2_struct_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc4_struct_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc37_membered,fc38_membered,fc39_membered,fc40_membered,fc41_membered,fc6_membered,rc1_membered,t1_boole,t1_subset,t3_boole,t4_boole,t4_subset,t5_subset,reflexivity_r1_tarski,existence_l2_rlvect_1,dt_l2_rlvect_1,cc15_membered,cc1_finset_1,cc1_funct_1,fc2_xboole_0,fc3_xboole_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_subset_1,redefinition_k6_subset_1,dt_k1_zfmisc_1,dt_k6_subset_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_subset_1,cc2_finset_1,fc12_finset_1,fc1_struct_0,fc9_finset_1,rc3_struct_0,t3_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,redefinition_k4_subset_1,dt_k2_xboole_0,dt_k4_subset_1,dt_k4_xboole_0,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,de_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,t39_xboole_1]), [interesting(0.02),file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_1_1__bhsp_7),[file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_1_1__bhsp_7)]]). fof(e7_4_2_4__bhsp_7,plain, ( ~ v1_xboole_0(c5_4_2_4__bhsp_7) & k8_funct_2(k5_numbers,k1_zfmisc_1(c2_4__bhsp_7),c1_4_2__bhsp_7,c2_4_2_4__bhsp_7) = c5_4_2_4__bhsp_7 & k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c2_4_2_4__bhsp_7) = k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c5_4_2_4__bhsp_7,c3_4__bhsp_7) ), inference(consider,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c5_4_2_4__bhsp_7,e6_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,e7_4_2_4__bhsp_7),[file(bhsp_7,e7_4_2_4__bhsp_7)]]). fof(e13_4_2_4__bhsp_7,plain,( r1_tarski(c5_4_2_4__bhsp_7,c7_4_2_4__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c2_4_2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e4_4_2__bhsp_7,e5_4_2_4__bhsp_7,e7_4_2_4__bhsp_7,e11_4_2_4__bhsp_7,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(bhsp_7,e13_4_2_4__bhsp_7),[file(bhsp_7,e13_4_2_4__bhsp_7)]]). fof(t12_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k2_xboole_0(A,B) = B ) ), file(xboole_1,t12_xboole_1), [interesting(0.9),axiom,file(xboole_1,t12_xboole_1)]). fof(e2_4_2_4_1_1_1_1_1_2_1_1__bhsp_7,plain,( k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c7_4_2_4__bhsp_7) = c7_4_2_4__bhsp_7 ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_struct_0,dt_l2_struct_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc4_struct_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc6_membered,rc1_membered,t1_boole,t1_subset,t4_subset,t5_subset,existence_l2_rlvect_1,dt_l2_rlvect_1,cc15_membered,cc1_finset_1,cc1_funct_1,fc2_xboole_0,fc3_xboole_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_subset_1,cc2_finset_1,fc1_struct_0,fc9_finset_1,rc3_struct_0,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,redefinition_k4_subset_1,dt_k2_xboole_0,dt_k4_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,t3_subset,e13_4_2_4__bhsp_7,t12_xboole_1]), [interesting(0.02),file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_1_1__bhsp_7),[file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_1_1__bhsp_7)]]). fof(e2_4_2_4_1_1_1_1_1_2_1__bhsp_7,plain,( k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7) = c7_4_2_4__bhsp_7 ), inference(iterative_eq,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[e1_4_2_4_1_1_1_1_1_2_1_1__bhsp_7,e2_4_2_4_1_1_1_1_1_2_1_1__bhsp_7]), [interesting(0.02),file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_1__bhsp_7),[file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_1__bhsp_7)]]). fof(t36_xboole_1,theorem,( ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ), file(xboole_1,t36_xboole_1), [interesting(0.9),axiom,file(xboole_1,t36_xboole_1)]). fof(e1_4_2_4_1_1_1_1_1_2_1__bhsp_7,plain,( r1_tarski(k6_subset_1(u1_struct_0(c1_4__bhsp_7),c7_4_2_4__bhsp_7,c5_4_2_4__bhsp_7),c7_4_2_4__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_struct_0,dt_l2_struct_0,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc4_struct_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_xboole_0,fc37_membered,fc38_membered,fc39_membered,fc40_membered,fc41_membered,fc6_membered,rc1_membered,t1_subset,t3_boole,t4_boole,t4_subset,t5_subset,existence_l2_rlvect_1,dt_l2_rlvect_1,cc15_membered,cc1_finset_1,cc1_funct_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_subset_1,cc2_finset_1,fc12_finset_1,fc1_struct_0,rc3_struct_0,reflexivity_r1_tarski,redefinition_k6_subset_1,dt_k4_xboole_0,dt_k6_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,t3_subset,t36_xboole_1]), [interesting(0.02),file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_1__bhsp_7),[file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_1__bhsp_7)]]). fof(t79_xboole_1,theorem,( ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) ), file(xboole_1,t79_xboole_1), [interesting(0.9),axiom,file(xboole_1,t79_xboole_1)]). fof(e3_4_2_4_1_1_1_1_1_2_1__bhsp_7,plain, ( ~ v1_xboole_0(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7) & r1_tarski(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7) = c7_4_2_4__bhsp_7 ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,rc1_funct_1,rc1_funct_2,rc1_xreal_0,rc2_funct_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc12_finset_1,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_xboole_0,fc37_membered,fc38_membered,fc39_membered,fc3_xboole_0,fc40_membered,fc41_membered,fc5_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t1_boole,t1_subset,t2_subset,t3_boole,t4_boole,t4_subset,t5_subset,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k6_subset_1,redefinition_k8_funct_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k4_subset_1,dt_k4_xboole_0,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_subset_1,dt_k8_funct_2,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c3_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,de_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_xboole_0,rc2_xboole_0,t3_subset,t6_boole,t7_boole,t8_boole,e2_4_2_4_1_1_1_1_1_2_1__bhsp_7,e11_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,e1_4_2_4_1_1_1_1_1_2_1__bhsp_7,t1_xboole_1,t79_xboole_1]), [interesting(0.02),file(bhsp_7,e3_4_2_4_1_1_1_1_1_2_1__bhsp_7),[file(bhsp_7,e3_4_2_4_1_1_1_1_1_2_1__bhsp_7)]]). fof(i2_4_2_4_1_1_1_1_1_2_1__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i2_4_2_4_1_1_1_1_1_2_1__bhsp_7)]), [interesting(0.02),trivial,file(bhsp_7,i2_4_2_4_1_1_1_1_1_2_1__bhsp_7)]). fof(i1_4_2_4_1_1_1_1_1_2_1__bhsp_7,plain, ( ~ v1_xboole_0(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7) & r1_tarski(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c1_4_2_4_1_1_1_1_1_2_1__bhsp_7) = c7_4_2_4__bhsp_7 ), inference(conclusion,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[e3_4_2_4_1_1_1_1_1_2_1__bhsp_7,i2_4_2_4_1_1_1_1_1_2_1__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_4_1_1_1_1_1_2_1__bhsp_7),[file(bhsp_7,i1_4_2_4_1_1_1_1_1_2_1__bhsp_7)]]). fof(e2_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(A,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,A) = c7_4_2_4__bhsp_7 ) ), inference(take,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,dt_l2_rlvect_1,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,commutativity_k2_xboole_0,idempotence_k2_xboole_0,dt_k2_xboole_0,dt_l1_bhsp_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_struct_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_xboole_0,fc3_xboole_0,fc9_finset_1,rc1_membered,rc3_struct_0,rc5_struct_0,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,redefinition_k4_subset_1,dt_k1_zfmisc_1,dt_k4_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2_1__bhsp_7,dt_c2_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,i1_4_2_4_1_1_1_1_1_2_1__bhsp_7]), [interesting(0.02),file(bhsp_7,e2_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e2_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e3_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ? [A] : ( v1_finset_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) & ~ v1_xboole_0(A) & r1_tarski(A,c2_4__bhsp_7) & r1_xboole_0(A,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,A) = c7_4_2_4__bhsp_7 ) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_l2_rlvect_1,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,dt_k1_xboole_0,dt_k2_xboole_0,dt_l1_bhsp_1,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc4_membered,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_xboole_0,fc3_xboole_0,fc6_membered,fc9_finset_1,rc1_membered,rc3_struct_0,rc5_struct_0,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,existence_m1_subset_1,redefinition_k4_subset_1,dt_k1_zfmisc_1,dt_k4_subset_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,e2_4_2_4_1_1_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,e3_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e3_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,plain, ( v1_finset_1(c1_4_2_4_1_1_1_1_1_2__bhsp_7) & m1_subset_1(c1_4_2_4_1_1_1_1_1_2__bhsp_7,k1_zfmisc_1(u1_struct_0(c1_4__bhsp_7))) ), inference(consider,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c1_4_2_4_1_1_1_1_1_2__bhsp_7,e3_4_2_4_1_1_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,c1_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,c1_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(t218_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,k7_xcmplx_0(A,2)) ) ) ), file(xreal_1,t218_xreal_1), [interesting(0.9),axiom,file(xreal_1,t218_xreal_1)]). fof(e10_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ~ r1_xreal_0(c1_4_2_4__bhsp_7,k7_xcmplx_0(c1_4_2_4__bhsp_7,2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7])],[reflexivity_r1_tarski,cc1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_membered,fc30_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc4_arithm,spc7_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k3_xcmplx_0,dt_k7_xcmplx_0,dt_c1_4_2_4__bhsp_7,cc2_xreal_0,fc4_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rn1d2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r0,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rn1d2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r0_r0,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e1_4_2_4__bhsp_7,t218_xreal_1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.02),file(bhsp_7,e10_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e10_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e4_4_2_4_1_1_1_1_1_2__bhsp_7,plain, ( ~ v1_xboole_0(c1_4_2_4_1_1_1_1_1_2__bhsp_7) & r1_tarski(c1_4_2_4_1_1_1_1_1_2__bhsp_7,c2_4__bhsp_7) & r1_xboole_0(c1_4_2_4_1_1_1_1_1_2__bhsp_7,c5_4_2_4__bhsp_7) & k4_subset_1(u1_struct_0(c1_4__bhsp_7),c5_4_2_4__bhsp_7,c1_4_2_4_1_1_1_1_1_2__bhsp_7) = c7_4_2_4__bhsp_7 ), inference(consider,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[dh_c1_4_2_4_1_1_1_1_1_2__bhsp_7,e3_4_2_4_1_1_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,e4_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e4_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e7_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ~ r1_xreal_0(k6_real_1(1,k1_nat_1(c2_4_2_4__bhsp_7,1)),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,rc2_finset_1,rc3_funct_1,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_xreal_0,cc7_xreal_0,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_xboole_0,fc30_xreal_0,fc3_xboole_0,fc3_xreal_0,fc5_membered,fc6_membered,fc6_xreal_0,fc8_xreal_0,fc9_finset_1,rc1_membered,rc1_xreal_0,rc3_struct_0,rc5_struct_0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc6_arithm,t1_boole,t1_real,t1_subset,t4_real,t4_subset,t5_subset,t6_arithm,projectivity_k18_complex1,commutativity_k1_nat_1,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k33_binop_2,dt_k4_subset_1,dt_k5_bhsp_5,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c2_4_2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc6_membered,cc9_membered,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,spc1_boole,spc1_numerals,e4_4_2__bhsp_7,e7_4_2_4__bhsp_7,e4_4_2_4_1_1_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,e7_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e7_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e4_4_2_4__bhsp_7,plain,( ~ r1_xreal_0(k7_xcmplx_0(c1_4_2_4__bhsp_7,2),k6_real_1(1,k1_nat_1(c2_4_2_4__bhsp_7,1))) ), inference(consider,[status(thm),assumptions([dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7])],[cc1_xreal_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,commutativity_k2_xcmplx_0,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_xreal_0,fc8_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_k6_real_1,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_4_2_4__bhsp_7,dt_c2_4_2_4__bhsp_7,fc2_membered,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,dh_c2_4_2_4__bhsp_7,e3_4_2_4__bhsp_7]), [interesting(0.5),file(bhsp_7,e4_4_2_4__bhsp_7),[file(bhsp_7,e4_4_2_4__bhsp_7)]]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]). fof(e8_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ~ r1_xreal_0(k7_xcmplx_0(c1_4_2_4__bhsp_7,2),k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc1_struct_0,fc23_xreal_0,fc30_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_finset_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k18_complex1,dt_k1_nat_1,dt_k1_numbers,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k5_bhsp_5,dt_k6_real_1,dt_k7_xcmplx_0,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c2_4_2_4__bhsp_7,dt_c3_4__bhsp_7,cc2_xreal_0,fc12_binop_2,fc2_membered,fc3_binop_2,fc4_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e7_4_2_4_1_1_1_1_1_2__bhsp_7,e4_4_2_4__bhsp_7,t2_xreal_1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r2_r1]), [interesting(0.02),file(bhsp_7,e8_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e8_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(involutiveness_k1_real_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), [interesting(0.9),axiom,file(real_1,k1_real_1)]). fof(redefinition_k1_real_1,definition,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), [interesting(0.9),axiom,file(real_1,k1_real_1)]). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), [interesting(0.9),axiom,file(real_1,k1_real_1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rnm1d2_rnm1d2,theorem,( k2_xcmplx_0(0,k7_xcmplx_0(k4_xcmplx_0(1),2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rnm1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rnm1d2_rnm1d2)]). fof(rqRealAdd__k2_xcmplx_0__r1_rnm1d2_rn1d2,theorem,( k2_xcmplx_0(1,k7_xcmplx_0(k4_xcmplx_0(1),2)) = k7_xcmplx_0(1,2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rnm1d2_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rnm1d2_rn1d2)]). fof(rqRealAdd__k2_xcmplx_0__rm1_rn1d2_rnm1d2,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),k7_xcmplx_0(1,2)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rn1d2_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rn1d2_rnm1d2)]). fof(rqRealAdd__k2_xcmplx_0__rn1d2_rm1_rnm1d2,theorem,( k2_xcmplx_0(k7_xcmplx_0(1,2),k4_xcmplx_0(1)) = k7_xcmplx_0(k4_xcmplx_0(1),2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_rm1_rnm1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_rm1_rnm1d2)]). fof(rqRealAdd__k2_xcmplx_0__rn1d2_rnm1d2_r0,theorem,( k2_xcmplx_0(k7_xcmplx_0(1,2),k7_xcmplx_0(k4_xcmplx_0(1),2)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_rnm1d2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rn1d2_rnm1d2_r0)]). fof(rqRealAdd__k2_xcmplx_0__rnm1d2_r1_rn1d2,theorem,( k2_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),1) = k7_xcmplx_0(1,2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rnm1d2_r1_rn1d2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rnm1d2_r1_rn1d2)]). fof(rqRealAdd__k2_xcmplx_0__rnm1d2_rn1d2_r0,theorem,( k2_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k7_xcmplx_0(1,2)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rnm1d2_rn1d2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rnm1d2_rn1d2_r0)]). fof(rqRealAdd__k2_xcmplx_0__rnm1d2_rnm1d2_rm1,theorem,( k2_xcmplx_0(k7_xcmplx_0(k4_xcmplx_0(1),2),k7_xcmplx_0(k4_xcmplx_0(1),2)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rnm1d2_rnm1d2_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rnm1d2_rnm1d2_rm1)]). fof(e5_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( k1_relat_1(c3_4__bhsp_7) = u1_struct_0(c1_4__bhsp_7) ), inference(mizar_by,[status(thm),assumptions([dt_c1_4__bhsp_7,dt_c3_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_numbers,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc6_membered,cc7_xreal_0,fc14_finset_1,fc2_membered,rc1_finset_1,rc1_xreal_0,rc3_finset_1,rc3_funct_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,redefinition_k6_supinf_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_relset_1,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc1_struct_0,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xboole_0,rc2_funct_1,rc2_xboole_0,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k4_relset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c3_4__bhsp_7,fc1_xboole_0,fc6_membered,t6_boole,d1_funct_2]), [interesting(0.02),file(bhsp_7,e5_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e5_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e1_4_2_4_1_1_1_1_1_2_2__bhsp_7,plain,( k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c7_4_2_4__bhsp_7,c3_4__bhsp_7) = k2_binop_1(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c5_4_2_4__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,rc3_funct_1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k1_binop_1,dt_k1_xboole_0,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,fc1_struct_0,fc1_xboole_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc3_struct_0,rc5_struct_0,t1_boole,t1_subset,t4_subset,t5_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,symmetry_r1_xboole_0,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_binop_1,redefinition_k4_subset_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_binop_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k4_subset_1,dt_k5_bhsp_5,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc6_membered,fc12_binop_2,fc14_finset_1,fc2_membered,fc2_xboole_0,fc3_binop_2,fc3_xboole_0,fc9_finset_1,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc4_finset_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,e5_4_2_4_1_1_1_1_1_2__bhsp_7,e4_4_2_4_1_1_1_1_1_2__bhsp_7,t14_bhsp_5]), [interesting(0.02),file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_2__bhsp_7),[file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_2__bhsp_7)]]). fof(e2_4_2_4_1_1_1_1_1_2_2__bhsp_7,plain,( k2_binop_1(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c5_4_2_4__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7)) = k3_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c5_4_2_4__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc1_xboole_0,fc3_xreal_0,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,spc6_arithm,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,redefinition_k6_supinf_1,dt_k1_binop_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,fc14_finset_1,fc1_struct_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,commutativity_k9_binop_2,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_binop_1,redefinition_k3_real_1,redefinition_k9_binop_2,redefinition_m2_relset_1,dt_k1_numbers,dt_k2_binop_1,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k3_real_1,dt_k5_bhsp_5,dt_k9_binop_2,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c3_4__bhsp_7,dt_c5_4_2_4__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,d9_binop_2]), [interesting(0.02),file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_2__bhsp_7),[file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_2__bhsp_7)]]). fof(e6_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c7_4_2_4__bhsp_7,c3_4__bhsp_7) = k3_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c5_4_2_4__bhsp_7,c3_4__bhsp_7),k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7)) ), inference(iterative_eq,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[e1_4_2_4_1_1_1_1_1_2_2__bhsp_7,e2_4_2_4_1_1_1_1_1_2_2__bhsp_7]), [interesting(0.02),file(bhsp_7,e6_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e6_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e1_4_2_4_1_1_1_1_1_2_3__bhsp_7,plain,( k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7))) = k18_complex1(k1_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7))) ), inference(mizar_by,[status(thm),assumptions([e1_4_2_4_1_1_1__bhsp_7,dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,dt_c3_4__bhsp_7,e2_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc7_xreal_0,fc9_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,existence_l2_rlvect_1,dt_k3_supinf_1,dt_l2_rlvect_1,cc1_funct_2,cc1_xreal_0,cc2_funct_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc30_xreal_0,fc3_xreal_0,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,rc1_funct_1,rc1_funct_2,rc1_xreal_0,rc2_funct_1,projectivity_k16_complex1,antisymmetry_r2_hidden,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc14_finset_1,fc1_ordinal2,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_subset,t2_subset,t3_subset,t4_arithm,t4_subset,t5_arithm,t5_subset,t6_arithm,projectivity_k18_complex1,involutiveness_k1_real_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,redefinition_k18_complex1,redefinition_k1_real_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k8_funct_2,dt_k18_complex1,dt_k1_numbers,dt_k1_real_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k33_binop_2,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_numbers,dt_k5_real_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_k8_funct_2,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c2_4__bhsp_7,dt_c2_4_2__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c5_4_2_4__bhsp_7,dt_c6_4_2_4__bhsp_7,dt_c7_4_2_4__bhsp_7,cc15_membered,cc1_finset_1,cc1_funct_1,fc12_binop_2,fc2_membered,fc3_binop_2,rc1_xboole_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r0_rn1d2_rn1d2,rqRealAdd__k2_xcmplx_0__r0_rnm1d2_rnm1d2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r1_rnm1d2_rn1d2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm1_rn1d2_rnm1d2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealAdd__k2_xcmplx_0__rn1d2_r0_rn1d2,rqRealAdd__k2_xcmplx_0__rn1d2_rm1_rnm1d2,rqRealAdd__k2_xcmplx_0__rn1d2_rn1d2_r1,rqRealAdd__k2_xcmplx_0__rn1d2_rnm1d2_r0,rqRealAdd__k2_xcmplx_0__rnm1d2_r1_rn1d2,rqRealAdd__k2_xcmplx_0__rnm1d2_rn1d2_r0,rqRealAdd__k2_xcmplx_0__rnm1d2_rnm1d2_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r0_rn1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__r0_rnm1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealDiff__k6_xcmplx_0__rn1d2_r0_rn1d2,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rn1d2_r0,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,rqRealDiff__k6_xcmplx_0__rnm1d2_r0_rnm1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,rqRealDiff__k6_xcmplx_0__rnm1d2_rnm1d2_r0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r0_r2_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,spc0_boole,spc1_boole,spc2_boole,t6_boole,t7_boole,t8_boole,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e9_4_2_4__bhsp_7,e11_4_2_4__bhsp_7,e1_4_2_4_1_1_1__bhsp_7,e6_4_2_4_1_1_1_1_1_2__bhsp_7,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0]), [interesting(0.02),file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_3__bhsp_7),[file(bhsp_7,e1_4_2_4_1_1_1_1_1_2_3__bhsp_7)]]). fof(projectivity_k17_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k17_complex1(A)) = k17_complex1(A) ) ), file(complex1,k17_complex1), [interesting(0.9),axiom,file(complex1,k17_complex1)]). fof(redefinition_k17_complex1,definition,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(A) = k16_complex1(A) ) ), file(complex1,k17_complex1), [interesting(0.9),axiom,file(complex1,k17_complex1)]). fof(dt_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k17_complex1(A),k1_numbers) ) ), file(complex1,k17_complex1), [interesting(0.9),axiom,file(complex1,k17_complex1)]). fof(t138_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k4_xcmplx_0(A)) = k17_complex1(A) ) ), file(complex1,t138_complex1), [interesting(0.9),axiom,file(complex1,t138_complex1)]). fof(e2_4_2_4_1_1_1_1_1_2_3__bhsp_7,plain,( k18_complex1(k1_real_1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7))) = k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7)) ), inference(mizar_by,[status(thm),assumptions([dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e2_4__bhsp_7,dt_c3_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_funct_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_k5_ordinal2,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc1_ordinal2,fc1_xboole_0,fc1_xreal_0,fc5_membered,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc1_xreal_0,rc2_funct_1,t1_subset,t4_subset,t5_subset,projectivity_k16_complex1,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_numbers,dt_k6_supinf_1,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc9_membered,fc14_finset_1,fc1_struct_0,rc1_finset_1,rc1_xboole_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,projectivity_k17_complex1,projectivity_k18_complex1,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,redefinition_k17_complex1,redefinition_k18_complex1,redefinition_k1_real_1,dt_k17_complex1,dt_k18_complex1,dt_k1_numbers,dt_k1_real_1,dt_k33_binop_2,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c3_4__bhsp_7,fc12_binop_2,fc2_membered,fc3_binop_2,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_boole,spc1_numerals,t138_complex1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.02),file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_3__bhsp_7),[file(bhsp_7,e2_4_2_4_1_1_1_1_1_2_3__bhsp_7)]]). fof(e9_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7))) = k18_complex1(k5_bhsp_5(k1_numbers,u1_struct_0(c1_4__bhsp_7),k33_binop_2,c1_4_2_4_1_1_1_1_1_2__bhsp_7,c3_4__bhsp_7)) ), inference(iterative_eq,[status(thm),assumptions([e1_4_2_4_1_1_1__bhsp_7,dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e2_4__bhsp_7,dt_c3_4__bhsp_7])],[e1_4_2_4_1_1_1_1_1_2_3__bhsp_7,e2_4_2_4_1_1_1_1_1_2_3__bhsp_7]), [interesting(0.02),file(bhsp_7,e9_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e9_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(e11_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ~ r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ), inference(mizar_by,[status(thm),assumptions([e1_4_2_4_1_1_1__bhsp_7,dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e2_4__bhsp_7,dt_c3_4__bhsp_7])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,cc1_xreal_0,rc2_finset_1,rc3_funct_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k3_supinf_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_funct_2,cc1_membered,cc20_membered,cc2_funct_1,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc6_membered,rc1_funct_1,rc1_funct_2,rc1_membered,rc2_funct_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k6_supinf_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k6_supinf_1,dt_k6_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc1_relset_1,cc2_finset_1,cc3_xreal_0,cc4_membered,cc5_funct_2,cc6_funct_2,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc14_finset_1,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_struct_0,fc23_xreal_0,fc30_xreal_0,fc5_membered,fc5_xreal_0,rc1_finset_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rc3_finset_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r1_rn1d2_rn1d2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm1_rnm1d2_rnm1d2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rn1d2_r1_rnm1d2,rqRealDiff__k6_xcmplx_0__rn1d2_rnm1d2_r1,rqRealDiff__k6_xcmplx_0__rnm1d2_rm1_rn1d2,rqRealDiff__k6_xcmplx_0__rnm1d2_rn1d2_rm1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k8_funct_2,dt_k18_complex1,dt_k1_numbers,dt_k33_binop_2,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_bhsp_5,dt_k5_numbers,dt_k5_real_1,dt_k7_xcmplx_0,dt_k8_funct_2,dt_u1_struct_0,dt_c1_4__bhsp_7,dt_c1_4_2_4__bhsp_7,dt_c1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c2_4_2__bhsp_7,dt_c3_4__bhsp_7,dt_c3_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,cc2_xreal_0,fc12_binop_2,fc1_xreal_0,fc2_membered,fc3_binop_2,fc4_xreal_0,fc6_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r1_rn1d2,rqLessOrEqual__r1_xreal_0__r1_rnm1d2,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__r2_rn1d2,rqLessOrEqual__r1_xreal_0__r2_rnm1d2,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm1_rn1d2,rqLessOrEqual__r1_xreal_0__rm1_rnm1d2,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqLessOrEqual__r1_xreal_0__rn1d2_r1,rqLessOrEqual__r1_xreal_0__rn1d2_r2,rqLessOrEqual__r1_xreal_0__rn1d2_rm1,rqLessOrEqual__r1_xreal_0__rn1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rn1d2_rnm1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_r1,rqLessOrEqual__r1_xreal_0__rnm1d2_r2,rqLessOrEqual__r1_xreal_0__rnm1d2_rm1,rqLessOrEqual__r1_xreal_0__rnm1d2_rn1d2,rqLessOrEqual__r1_xreal_0__rnm1d2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm2_rnm1d2,rqRealDiv__k7_xcmplx_0__r1_rn1d2_r2,rqRealDiv__k7_xcmplx_0__r1_rnm1d2_rm2,rqRealDiv__k7_xcmplx_0__r2_r1_r2,rqRealDiv__k7_xcmplx_0__r2_r2_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r2_rnm1d2,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r1_rn1d2_rn1d2,rqRealMult__k3_xcmplx_0__r1_rnm1d2_rnm1d2,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__r2_rn1d2_r1,rqRealMult__k3_xcmplx_0__r2_rnm1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealMult__k3_xcmplx_0__rm2_rn1d2_rm1,rqRealMult__k3_xcmplx_0__rm2_rnm1d2_r1,rqRealMult__k3_xcmplx_0__rn1d2_r2_r1,rqRealMult__k3_xcmplx_0__rn1d2_rm2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_r1_rnm1d2,rqRealMult__k3_xcmplx_0__rnm1d2_r2_rm1,rqRealMult__k3_xcmplx_0__rnm1d2_rm2_r1,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,rqRealNeg__k4_xcmplx_0__rn1d2_rnm1d2,rqRealNeg__k4_xcmplx_0__rnm1d2_rn1d2,spc1_boole,spc2_boole,spc1_numerals,spc2_numerals,e10_4_2_4_1_1_1_1_1_2__bhsp_7,e8_4_2_4_1_1_1_1_1_2__bhsp_7,e9_4_2_4_1_1_1_1_1_2__bhsp_7,t2_xreal_1,rqRealDiv__k7_xcmplx_0__r1_r2_rn1d2,rqRealMult__k3_xcmplx_0__rn1d2_r1_rn1d2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.02),file(bhsp_7,e11_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,e11_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(i2_4_2_4_1_1_1_1_1_2__bhsp_7,theorem,( $true ), introduced(tautology,[file(bhsp_7,i2_4_2_4_1_1_1_1_1_2__bhsp_7)]), [interesting(0.02),trivial,file(bhsp_7,i2_4_2_4_1_1_1_1_1_2__bhsp_7)]). fof(i1_4_2_4_1_1_1_1_1_2__bhsp_7,plain,( ~ r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ), inference(conclusion,[status(thm),assumptions([e1_4_2_4_1_1_1__bhsp_7,dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,e1_4_2_4_1_1_1_1_1_2__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e2_4__bhsp_7,dt_c3_4__bhsp_7])],[e11_4_2_4_1_1_1_1_1_2__bhsp_7,i2_4_2_4_1_1_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,i1_4_2_4_1_1_1_1_1_2__bhsp_7),[file(bhsp_7,i1_4_2_4_1_1_1_1_1_2__bhsp_7)]]). fof(i2_4_2_4_1_1_1_1_1__bhsp_7,plain, ( c5_4_2_4__bhsp_7 != c7_4_2_4__bhsp_7 => ( c5_4_2_4__bhsp_7 != c7_4_2_4__bhsp_7 & ~ r1_xreal_0(c1_4_2_4__bhsp_7,k18_complex1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c3_4_2_4__bhsp_7),k8_funct_2(k5_numbers,k1_numbers,c2_4_2__bhsp_7,c4_4_2_4__bhsp_7)))) ) ), inference(discharge_asm,[status(thm),assumptions([e1_4_2_4_1_1_1__bhsp_7,dt_c3_4_2_4__bhsp_7,e5_4_2_4__bhsp_7,dt_c1_4_2_4__bhsp_7,e1_4_2_4__bhsp_7,dt_c4_4_2_4__bhsp_7,dt_c1_4__bhsp_7,dt_c2_4__bhsp_7,e2_4__bhsp_7,dt_c3_4__bhsp_7]),discharge_asm(discharge,[e1_4_2_4_1_1_1_1_1_2__bhsp_7])],[e1_4_2_4_1_1_1_1_1_2__bhsp_7,i1_4_2_4_1_1_1_1_1_2__bhsp_7]), [interesting(0.02),file(bhsp_7,i2_4_2_4_1_1_1_1_1__bhsp_7),[file(bhsp_7,i2_4_2_4_1_1_1