% Mizar ND problem: t9_bhsp_2,bhsp_2,258,28 fof(dh_c1_10__bhsp_2,definition, ( ( ( ~ v3_struct_0(c1_10__bhsp_2) & v3_rlvect_1(c1_10__bhsp_2) & v4_rlvect_1(c1_10__bhsp_2) & v5_rlvect_1(c1_10__bhsp_2) & v6_rlvect_1(c1_10__bhsp_2) & v7_rlvect_1(c1_10__bhsp_2) & v2_bhsp_1(c1_10__bhsp_2) & l1_bhsp_1(c1_10__bhsp_2) ) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ) => ( v1_bhsp_2(A,c1_10__bhsp_2) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(c1_10__bhsp_2)) & ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,A,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] : ( ( v1_funct_1(G) & v1_funct_2(G,k5_numbers,u1_struct_0(F)) & m2_relset_1(G,k5_numbers,u1_struct_0(F)) ) => ( v1_bhsp_2(G,F) <=> ? [H] : ( m1_subset_1(H,u1_struct_0(F)) & ! [I] : ( m1_subset_1(I,k1_numbers) => ~ ( ~ r1_xreal_0(I,0) & ! [J] : ( m2_subset_1(J,k1_numbers,k5_numbers) => ? [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) & r1_xreal_0(J,K) & r1_xreal_0(I,k3_bhsp_1(F,k6_rlvect_1(F,k2_normsp_1(F,G,K),H))) ) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_10__bhsp_2),file(bhsp_2,c1_10__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c1_10__bhsp_2)]). fof(dh_c2_10__bhsp_2,definition, ( ( ( v1_funct_1(c2_10__bhsp_2) & v1_funct_2(c2_10__bhsp_2,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(c2_10__bhsp_2,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ) => ( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) <=> ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(E,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ) => ( v1_bhsp_2(E,c1_10__bhsp_2) <=> ? [F] : ( m1_subset_1(F,u1_struct_0(c1_10__bhsp_2)) & ! [G] : ( m1_subset_1(G,k1_numbers) => ~ ( ~ r1_xreal_0(G,0) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ? [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) & r1_xreal_0(H,I) & r1_xreal_0(G,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,E,I),F))) ) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_10__bhsp_2),file(bhsp_2,c2_10__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c2_10__bhsp_2)]). 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_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(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(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(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_arytm_3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). 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(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(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_arytm_3,theorem,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). 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(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). 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(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_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(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(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(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). 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_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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_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(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(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(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(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(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_arytm_3,theorem,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). 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(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(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(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(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(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(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(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(redefinition_k2_normsp_1,definition,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) ), file(normsp_1,k2_normsp_1), [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). 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_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k2_normsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) ), file(normsp_1,k2_normsp_1), [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]). fof(dt_k3_bhsp_1,axiom,( ! [A,B] : ( ( ~ 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,u1_struct_0(A)) ) => m1_subset_1(k3_bhsp_1(A,B),k1_numbers) ) ), file(bhsp_1,k3_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k3_bhsp_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_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k6_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k6_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k6_rlvect_1)]). 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_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(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_10__bhsp_2,assumption, ( ~ v3_struct_0(c1_10__bhsp_2) & v3_rlvect_1(c1_10__bhsp_2) & v4_rlvect_1(c1_10__bhsp_2) & v5_rlvect_1(c1_10__bhsp_2) & v6_rlvect_1(c1_10__bhsp_2) & v7_rlvect_1(c1_10__bhsp_2) & v2_bhsp_1(c1_10__bhsp_2) & l1_bhsp_1(c1_10__bhsp_2) ), introduced(assumption,[file(bhsp_2,c1_10__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c1_10__bhsp_2)]). fof(dt_c2_10__bhsp_2,assumption, ( v1_funct_1(c2_10__bhsp_2) & v1_funct_2(c2_10__bhsp_2,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(c2_10__bhsp_2,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ), introduced(assumption,[file(bhsp_2,c2_10__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c2_10__bhsp_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(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(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(e1_10_1__bhsp_2,assumption,( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) ), introduced(assumption,[file(bhsp_2,e1_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e1_10_1__bhsp_2)]). fof(dt_k4_bhsp_1,axiom,( ! [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,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_bhsp_1(A,B,C),k1_numbers) ) ), file(bhsp_1,k4_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k4_bhsp_1)]). fof(commutativity_k5_bhsp_1,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,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k5_bhsp_1(A,B,C) = k5_bhsp_1(A,C,B) ) ), file(bhsp_1,k5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k5_bhsp_1)]). fof(redefinition_k5_bhsp_1,definition,( ! [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,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k5_bhsp_1(A,B,C) = k4_bhsp_1(A,B,C) ) ), file(bhsp_1,k5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k5_bhsp_1)]). fof(dt_k5_bhsp_1,axiom,( ! [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,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k5_bhsp_1(A,B,C),k1_numbers) ) ), file(bhsp_1,k5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,k5_bhsp_1)]). fof(dh_c1_10_1__bhsp_2,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A)) ) ) ) ) ) => ( m1_subset_1(c1_10_1__bhsp_2,u1_struct_0(c1_10__bhsp_2)) & ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( ~ 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,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,G),c1_10_1__bhsp_2)) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_10_1__bhsp_2),file(bhsp_2,c1_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c1_10_1__bhsp_2)]). 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(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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_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_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_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(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(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(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_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(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(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(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_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(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(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(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(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(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(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(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_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(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(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(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(d1_bhsp_2,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] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( v1_bhsp_2(B,A) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(A)) & ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & r1_xreal_0(E,F) & r1_xreal_0(D,k5_bhsp_1(A,k2_normsp_1(A,B,F),C)) ) ) ) ) ) ) ) ) ), file(bhsp_2,d1_bhsp_2), [interesting(0.9),axiom,file(bhsp_2,d1_bhsp_2)]). fof(e2_10_1__bhsp_2,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A)) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_l1_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,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,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e1_10_1__bhsp_2,d1_bhsp_2]), [interesting(0.65),file(bhsp_2,e2_10_1__bhsp_2),[file(bhsp_2,e2_10_1__bhsp_2)]]). fof(dt_c1_10_1__bhsp_2,plain,( m1_subset_1(c1_10_1__bhsp_2,u1_struct_0(c1_10__bhsp_2)) ), inference(consider,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,dh_c1_10_1__bhsp_2,e2_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,c1_10_1__bhsp_2),[file(bhsp_2,c1_10_1__bhsp_2)]]). fof(dh_c2_10_1__bhsp_2,definition, ( ( m1_subset_1(c2_10_1__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c2_10_1__bhsp_2,0) & ! [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(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_1__bhsp_2))) ) ) ) ) => ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,E),c1_10_1__bhsp_2))) ) ) ) ) ), introduced(definition,[new_symbol(c2_10_1__bhsp_2),file(bhsp_2,c2_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c2_10_1__bhsp_2)]). fof(e4_10_1__bhsp_2,assumption,( ~ r1_xreal_0(c2_10_1__bhsp_2,0) ), introduced(assumption,[file(bhsp_2,e4_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e4_10_1__bhsp_2)]). fof(dt_c2_10_1__bhsp_2,assumption,( m1_subset_1(c2_10_1__bhsp_2,k1_numbers) ), introduced(assumption,[file(bhsp_2,c2_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c2_10_1__bhsp_2)]). fof(dh_c3_10_1__bhsp_2,definition, ( ? [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(c2_10_1__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_1__bhsp_2)) ) ) ) => ( m2_subset_1(c3_10_1__bhsp_2,k1_numbers,k5_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_10_1__bhsp_2,C) & r1_xreal_0(c2_10_1__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,C),c1_10_1__bhsp_2)) ) ) ) ), introduced(definition,[new_symbol(c3_10_1__bhsp_2),file(bhsp_2,c3_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c3_10_1__bhsp_2)]). fof(e3_10_1__bhsp_2,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(B,C) & r1_xreal_0(A,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,C),c1_10_1__bhsp_2)) ) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c1_10_1__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,dh_c1_10_1__bhsp_2,e2_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e3_10_1__bhsp_2),[file(bhsp_2,e3_10_1__bhsp_2)]]). fof(e5_10_1__bhsp_2,plain,( ? [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(c2_10_1__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_1__bhsp_2)) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,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,existence_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_10__bhsp_2,dt_c1_10_1__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_1__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e4_10_1__bhsp_2,e3_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e5_10_1__bhsp_2),[file(bhsp_2,e5_10_1__bhsp_2)]]). fof(dt_c3_10_1__bhsp_2,plain,( m2_subset_1(c3_10_1__bhsp_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[dh_c3_10_1__bhsp_2,e5_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,c3_10_1__bhsp_2),[file(bhsp_2,c3_10_1__bhsp_2)]]). fof(de_c4_10_1__bhsp_2,definition,( c4_10_1__bhsp_2 = c3_10_1__bhsp_2 ), introduced(definition,[new_symbol(c4_10_1__bhsp_2),file(bhsp_2,c4_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c4_10_1__bhsp_2)]). fof(dt_c4_10_1__bhsp_2,plain,( m2_subset_1(c4_10_1__bhsp_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[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_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,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,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,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_c3_10_1__bhsp_2,fc2_membered,de_c4_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,c4_10_1__bhsp_2),[file(bhsp_2,c4_10_1__bhsp_2)]]). fof(dh_c5_10_1__bhsp_2,definition, ( ( m2_subset_1(c5_10_1__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_1__bhsp_2,c5_10_1__bhsp_2) & r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_1__bhsp_2),c1_10_1__bhsp_2))) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_1__bhsp_2,A) & r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,A),c1_10_1__bhsp_2))) ) ) ), introduced(definition,[new_symbol(c5_10_1__bhsp_2),file(bhsp_2,c5_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c5_10_1__bhsp_2)]). fof(e7_10_1__bhsp_2,assumption,( r1_xreal_0(c4_10_1__bhsp_2,c5_10_1__bhsp_2) ), introduced(assumption,[file(bhsp_2,e7_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e7_10_1__bhsp_2)]). fof(dt_c5_10_1__bhsp_2,assumption,( m2_subset_1(c5_10_1__bhsp_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_2,c5_10_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c5_10_1__bhsp_2)]). fof(e6_10_1__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_10_1__bhsp_2,A) & r1_xreal_0(c2_10_1__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,A),c1_10_1__bhsp_2)) ) ) ), inference(consider,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[dh_c3_10_1__bhsp_2,e5_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e6_10_1__bhsp_2),[file(bhsp_2,e6_10_1__bhsp_2)]]). fof(e8_10_1__bhsp_2,plain,( ~ r1_xreal_0(c2_10_1__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_1__bhsp_2),c1_10_1__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c5_10_1__bhsp_2,e7_10_1__bhsp_2,dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[existence_l2_struct_0,dt_l2_struct_0,rc4_struct_0,existence_l1_rlvect_1,dt_l1_rlvect_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,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_l1_bhsp_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_10__bhsp_2,dt_c1_10_1__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_1__bhsp_2,dt_c3_10_1__bhsp_2,dt_c4_10_1__bhsp_2,dt_c5_10_1__bhsp_2,de_c4_10_1__bhsp_2,fc2_membered,e7_10_1__bhsp_2,e6_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e8_10_1__bhsp_2),[file(bhsp_2,e8_10_1__bhsp_2)]]). fof(d5_bhsp_1,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,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k4_bhsp_1(A,B,C) = k3_bhsp_1(A,k6_rlvect_1(A,B,C)) ) ) ) ), file(bhsp_1,d5_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,d5_bhsp_1)]). fof(e9_10_1__bhsp_2,plain,( ~ r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_1__bhsp_2),c1_10_1__bhsp_2))) ), inference(mizar_by,[status(thm),assumptions([dt_c5_10_1__bhsp_2,e7_10_1__bhsp_2,dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_xreal_0,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_l2_struct_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xreal_0,rc4_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc2_xreal_0,fc1_struct_0,fc2_membered,rc3_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k4_bhsp_1,dt_k5_bhsp_1,dt_k6_rlvect_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c1_10_1__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_1__bhsp_2,dt_c5_10_1__bhsp_2,e8_10_1__bhsp_2,d5_bhsp_1]), [interesting(0.65),file(bhsp_2,e9_10_1__bhsp_2),[file(bhsp_2,e9_10_1__bhsp_2)]]). fof(i8_10_1__bhsp_2,theorem,( $true ), introduced(tautology,[file(bhsp_2,i8_10_1__bhsp_2)]), [interesting(0.65),trivial,file(bhsp_2,i8_10_1__bhsp_2)]). fof(i7_10_1__bhsp_2,plain,( ~ r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_1__bhsp_2),c1_10_1__bhsp_2))) ), inference(conclusion,[status(thm),assumptions([dt_c5_10_1__bhsp_2,e7_10_1__bhsp_2,dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[e9_10_1__bhsp_2,i8_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i7_10_1__bhsp_2),[file(bhsp_2,i7_10_1__bhsp_2)]]). fof(i6_10_1__bhsp_2,plain,( ~ ( r1_xreal_0(c4_10_1__bhsp_2,c5_10_1__bhsp_2) & r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_1__bhsp_2),c1_10_1__bhsp_2))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c5_10_1__bhsp_2,dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2]),discharge_asm(discharge,[e7_10_1__bhsp_2])],[e7_10_1__bhsp_2,i7_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i6_10_1__bhsp_2),[file(bhsp_2,i6_10_1__bhsp_2)]]). fof(i6_10_1_tmp__bhsp_2,plain, ( m2_subset_1(c5_10_1__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_1__bhsp_2,c5_10_1__bhsp_2) & r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_1__bhsp_2),c1_10_1__bhsp_2))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2]),discharge_asm(discharge,[dt_c5_10_1__bhsp_2])],[dt_c5_10_1__bhsp_2,i6_10_1__bhsp_2]), [interesting(0.65),i5_10_1__bhsp_2]). fof(i5_10_1__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_1__bhsp_2,A) & r1_xreal_0(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,A),c1_10_1__bhsp_2))) ) ) ), inference(let,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[i6_10_1_tmp__bhsp_2,dh_c5_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i5_10_1__bhsp_2),[file(bhsp_2,i5_10_1__bhsp_2)]]). fof(i4_10_1__bhsp_2,plain,( ? [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(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_1__bhsp_2))) ) ) ) ), inference(take,[status(thm),assumptions([dt_c2_10_1__bhsp_2,e4_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m2_subset_1,dt_c1_10__bhsp_2,dt_c1_10_1__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_1__bhsp_2,dt_c4_10_1__bhsp_2,fc2_membered,i5_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i4_10_1__bhsp_2),[file(bhsp_2,i4_10_1__bhsp_2)]]). fof(i3_10_1__bhsp_2,plain,( ~ ( ~ r1_xreal_0(c2_10_1__bhsp_2,0) & ! [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(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_1__bhsp_2))) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_10_1__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2]),discharge_asm(discharge,[e4_10_1__bhsp_2])],[e4_10_1__bhsp_2,i4_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i3_10_1__bhsp_2),[file(bhsp_2,i3_10_1__bhsp_2)]]). fof(i3_10_1_tmp__bhsp_2,plain, ( m1_subset_1(c2_10_1__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c2_10_1__bhsp_2,0) & ! [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(c2_10_1__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_1__bhsp_2))) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2]),discharge_asm(discharge,[dt_c2_10_1__bhsp_2])],[dt_c2_10_1__bhsp_2,i3_10_1__bhsp_2]), [interesting(0.65),i2_10_1__bhsp_2]). fof(i2_10_1__bhsp_2,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(B,C) & r1_xreal_0(A,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,C),c1_10_1__bhsp_2))) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[i3_10_1_tmp__bhsp_2,dh_c2_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i2_10_1__bhsp_2),[file(bhsp_2,i2_10_1__bhsp_2)]]). fof(i1_10_1__bhsp_2,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ), inference(take,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_1__bhsp_2])],[dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c1_10_1__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,i2_10_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i1_10_1__bhsp_2),[file(bhsp_2,i1_10_1__bhsp_2)]]). fof(i1_10_1_tmp__bhsp_2,plain, ( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) => ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2]),discharge_asm(discharge,[e1_10_1__bhsp_2])],[e1_10_1__bhsp_2,i1_10_1__bhsp_2]), [interesting(0.8),e1_10__bhsp_2]). fof(e1_10__bhsp_2,plain,( ~ ( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) & ! [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) => ? [B] : ( m1_subset_1(B,k1_numbers) & ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2])],[i1_10_1_tmp__bhsp_2,dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole]), [interesting(0.8),file(bhsp_2,e1_10__bhsp_2),[file(bhsp_2,e1_10__bhsp_2)]]). fof(e1_10_2__bhsp_2,assumption,( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ), introduced(assumption,[file(bhsp_2,e1_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e1_10_2__bhsp_2)]). fof(dh_c1_10_2__bhsp_2,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) => ( m1_subset_1(c1_10_2__bhsp_2,u1_struct_0(c1_10__bhsp_2)) & ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,G),c1_10_2__bhsp_2))) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_10_2__bhsp_2),file(bhsp_2,c1_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c1_10_2__bhsp_2)]). fof(dt_c1_10_2__bhsp_2,plain,( m1_subset_1(c1_10_2__bhsp_2,u1_struct_0(c1_10__bhsp_2)) ), inference(consider,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,dh_c1_10_2__bhsp_2,e1_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,c1_10_2__bhsp_2),[file(bhsp_2,c1_10_2__bhsp_2)]]). fof(dh_c2_10_2__bhsp_2,definition, ( ( m1_subset_1(c2_10_2__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c2_10_2__bhsp_2,0) & ! [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(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_2__bhsp_2)) ) ) ) ) => ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ 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,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,E),c1_10_2__bhsp_2)) ) ) ) ) ), introduced(definition,[new_symbol(c2_10_2__bhsp_2),file(bhsp_2,c2_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c2_10_2__bhsp_2)]). fof(e3_10_2__bhsp_2,assumption,( ~ r1_xreal_0(c2_10_2__bhsp_2,0) ), introduced(assumption,[file(bhsp_2,e3_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e3_10_2__bhsp_2)]). fof(dt_c2_10_2__bhsp_2,assumption,( m1_subset_1(c2_10_2__bhsp_2,k1_numbers) ), introduced(assumption,[file(bhsp_2,c2_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c2_10_2__bhsp_2)]). fof(dh_c3_10_2__bhsp_2,definition, ( ? [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(c2_10_2__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_2__bhsp_2))) ) ) ) => ( m2_subset_1(c3_10_2__bhsp_2,k1_numbers,k5_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_10_2__bhsp_2,C) & r1_xreal_0(c2_10_2__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,C),c1_10_2__bhsp_2))) ) ) ) ), introduced(definition,[new_symbol(c3_10_2__bhsp_2),file(bhsp_2,c3_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c3_10_2__bhsp_2)]). fof(e2_10_2__bhsp_2,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(B,C) & r1_xreal_0(A,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,C),c1_10_2__bhsp_2))) ) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c1_10_2__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,dh_c1_10_2__bhsp_2,e1_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e2_10_2__bhsp_2),[file(bhsp_2,e2_10_2__bhsp_2)]]). fof(e4_10_2__bhsp_2,plain,( ? [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(c2_10_2__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_2__bhsp_2))) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,existence_l2_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_10__bhsp_2,dt_c1_10_2__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_2__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e3_10_2__bhsp_2,e2_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e4_10_2__bhsp_2),[file(bhsp_2,e4_10_2__bhsp_2)]]). fof(dt_c3_10_2__bhsp_2,plain,( m2_subset_1(c3_10_2__bhsp_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[dh_c3_10_2__bhsp_2,e4_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,c3_10_2__bhsp_2),[file(bhsp_2,c3_10_2__bhsp_2)]]). fof(de_c4_10_2__bhsp_2,definition,( c4_10_2__bhsp_2 = c3_10_2__bhsp_2 ), introduced(definition,[new_symbol(c4_10_2__bhsp_2),file(bhsp_2,c4_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c4_10_2__bhsp_2)]). fof(dt_c4_10_2__bhsp_2,plain,( m2_subset_1(c4_10_2__bhsp_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[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_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,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,cc3_arytm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,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_c3_10_2__bhsp_2,fc2_membered,de_c4_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,c4_10_2__bhsp_2),[file(bhsp_2,c4_10_2__bhsp_2)]]). fof(dh_c5_10_2__bhsp_2,definition, ( ( m2_subset_1(c5_10_2__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_2__bhsp_2,c5_10_2__bhsp_2) & r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_2__bhsp_2),c1_10_2__bhsp_2)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_2__bhsp_2,A) & r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,A),c1_10_2__bhsp_2)) ) ) ), introduced(definition,[new_symbol(c5_10_2__bhsp_2),file(bhsp_2,c5_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c5_10_2__bhsp_2)]). fof(e6_10_2__bhsp_2,assumption,( r1_xreal_0(c4_10_2__bhsp_2,c5_10_2__bhsp_2) ), introduced(assumption,[file(bhsp_2,e6_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e6_10_2__bhsp_2)]). fof(dt_c5_10_2__bhsp_2,assumption,( m2_subset_1(c5_10_2__bhsp_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_2,c5_10_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c5_10_2__bhsp_2)]). fof(e5_10_2__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_10_2__bhsp_2,A) & r1_xreal_0(c2_10_2__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,A),c1_10_2__bhsp_2))) ) ) ), inference(consider,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[dh_c3_10_2__bhsp_2,e4_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e5_10_2__bhsp_2),[file(bhsp_2,e5_10_2__bhsp_2)]]). fof(e7_10_2__bhsp_2,plain,( ~ r1_xreal_0(c2_10_2__bhsp_2,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_2__bhsp_2),c1_10_2__bhsp_2))) ), inference(mizar_by,[status(thm),assumptions([dt_c5_10_2__bhsp_2,e6_10_2__bhsp_2,dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,existence_l2_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,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_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m2_subset_1,dt_c1_10__bhsp_2,dt_c1_10_2__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_2__bhsp_2,dt_c3_10_2__bhsp_2,dt_c4_10_2__bhsp_2,dt_c5_10_2__bhsp_2,de_c4_10_2__bhsp_2,fc2_membered,e6_10_2__bhsp_2,e5_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e7_10_2__bhsp_2),[file(bhsp_2,e7_10_2__bhsp_2)]]). fof(e8_10_2__bhsp_2,plain,( ~ r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_2__bhsp_2),c1_10_2__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c5_10_2__bhsp_2,e6_10_2__bhsp_2,dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc2_xreal_0,rc3_xreal_0,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_l2_struct_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc1_xreal_0,rc4_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc2_xreal_0,fc1_struct_0,fc2_membered,rc3_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k4_bhsp_1,dt_k5_bhsp_1,dt_k6_rlvect_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c1_10_2__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_2__bhsp_2,dt_c5_10_2__bhsp_2,e7_10_2__bhsp_2,d5_bhsp_1]), [interesting(0.65),file(bhsp_2,e8_10_2__bhsp_2),[file(bhsp_2,e8_10_2__bhsp_2)]]). fof(i8_10_2__bhsp_2,theorem,( $true ), introduced(tautology,[file(bhsp_2,i8_10_2__bhsp_2)]), [interesting(0.65),trivial,file(bhsp_2,i8_10_2__bhsp_2)]). fof(i7_10_2__bhsp_2,plain,( ~ r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_2__bhsp_2),c1_10_2__bhsp_2)) ), inference(conclusion,[status(thm),assumptions([dt_c5_10_2__bhsp_2,e6_10_2__bhsp_2,dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[e8_10_2__bhsp_2,i8_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i7_10_2__bhsp_2),[file(bhsp_2,i7_10_2__bhsp_2)]]). fof(i6_10_2__bhsp_2,plain,( ~ ( r1_xreal_0(c4_10_2__bhsp_2,c5_10_2__bhsp_2) & r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_2__bhsp_2),c1_10_2__bhsp_2)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c5_10_2__bhsp_2,dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2]),discharge_asm(discharge,[e6_10_2__bhsp_2])],[e6_10_2__bhsp_2,i7_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i6_10_2__bhsp_2),[file(bhsp_2,i6_10_2__bhsp_2)]]). fof(i6_10_2_tmp__bhsp_2,plain, ( m2_subset_1(c5_10_2__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_2__bhsp_2,c5_10_2__bhsp_2) & r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,c5_10_2__bhsp_2),c1_10_2__bhsp_2)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2]),discharge_asm(discharge,[dt_c5_10_2__bhsp_2])],[dt_c5_10_2__bhsp_2,i6_10_2__bhsp_2]), [interesting(0.65),i5_10_2__bhsp_2]). fof(i5_10_2__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c4_10_2__bhsp_2,A) & r1_xreal_0(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,A),c1_10_2__bhsp_2)) ) ) ), inference(let,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[i6_10_2_tmp__bhsp_2,dh_c5_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i5_10_2__bhsp_2),[file(bhsp_2,i5_10_2__bhsp_2)]]). fof(i4_10_2__bhsp_2,plain,( ? [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(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_2__bhsp_2)) ) ) ) ), inference(take,[status(thm),assumptions([dt_c2_10_2__bhsp_2,e3_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,dt_k2_zfmisc_1,dt_l2_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_10__bhsp_2,dt_c1_10_2__bhsp_2,dt_c2_10__bhsp_2,dt_c2_10_2__bhsp_2,dt_c4_10_2__bhsp_2,fc2_membered,i5_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i4_10_2__bhsp_2),[file(bhsp_2,i4_10_2__bhsp_2)]]). fof(i3_10_2__bhsp_2,plain,( ~ ( ~ r1_xreal_0(c2_10_2__bhsp_2,0) & ! [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(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_2__bhsp_2)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_10_2__bhsp_2,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2]),discharge_asm(discharge,[e3_10_2__bhsp_2])],[e3_10_2__bhsp_2,i4_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i3_10_2__bhsp_2),[file(bhsp_2,i3_10_2__bhsp_2)]]). fof(i3_10_2_tmp__bhsp_2,plain, ( m1_subset_1(c2_10_2__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c2_10_2__bhsp_2,0) & ! [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(c2_10_2__bhsp_2,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,B),c1_10_2__bhsp_2)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2]),discharge_asm(discharge,[dt_c2_10_2__bhsp_2])],[dt_c2_10_2__bhsp_2,i3_10_2__bhsp_2]), [interesting(0.65),i2_10_2__bhsp_2]). fof(i2_10_2__bhsp_2,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(B,C) & r1_xreal_0(A,k5_bhsp_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,C),c1_10_2__bhsp_2)) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[i3_10_2_tmp__bhsp_2,dh_c2_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i2_10_2__bhsp_2),[file(bhsp_2,i2_10_2__bhsp_2)]]). fof(i1_10_2__bhsp_2,plain,( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) ), inference(take,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,e1_10_2__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_l1_rlvect_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c1_10_2__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,d1_bhsp_2,i2_10_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i1_10_2__bhsp_2),[file(bhsp_2,i1_10_2__bhsp_2)]]). fof(i1_10_2_tmp__bhsp_2,plain, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) => v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2]),discharge_asm(discharge,[e1_10_2__bhsp_2])],[e1_10_2__bhsp_2,i1_10_2__bhsp_2]), [interesting(0.8),e2_10__bhsp_2]). fof(e2_10__bhsp_2,plain, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) => v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2])],[i1_10_2_tmp__bhsp_2,dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole]), [interesting(0.8),file(bhsp_2,e2_10__bhsp_2),[file(bhsp_2,e2_10__bhsp_2)]]). fof(e3_10__bhsp_2,plain, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) => v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_rlvect_1,existence_l2_struct_0,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e2_10__bhsp_2]), [interesting(0.8),file(bhsp_2,e3_10__bhsp_2),[file(bhsp_2,e3_10__bhsp_2)]]). fof(i4_10__bhsp_2,theorem,( $true ), introduced(tautology,[file(bhsp_2,i4_10__bhsp_2)]), [interesting(0.8),trivial,file(bhsp_2,i4_10__bhsp_2)]). fof(i3_10__bhsp_2,plain, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) => v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) ), inference(conclusion,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2])],[dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,e3_10__bhsp_2,i4_10__bhsp_2]), [interesting(0.8),file(bhsp_2,i3_10__bhsp_2),[file(bhsp_2,i3_10__bhsp_2)]]). fof(i2_10__bhsp_2,plain, ( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) <=> ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_10__bhsp_2,dt_c2_10__bhsp_2])],[dt_k2_zfmisc_1,dt_l2_rlvect_1,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_struct_0,rc4_xreal_0,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k5_numbers,dt_k6_rlvect_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_10__bhsp_2,dt_c2_10__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,e1_10__bhsp_2,i3_10__bhsp_2]), [interesting(0.8),file(bhsp_2,i2_10__bhsp_2),[file(bhsp_2,i2_10__bhsp_2)]]). fof(i2_10_tmp__bhsp_2,plain, ( ( v1_funct_1(c2_10__bhsp_2) & v1_funct_2(c2_10__bhsp_2,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(c2_10__bhsp_2,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ) => ( v1_bhsp_2(c2_10__bhsp_2,c1_10__bhsp_2) <=> ? [A] : ( m1_subset_1(A,u1_struct_0(c1_10__bhsp_2)) & ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,c2_10__bhsp_2,D),A))) ) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_10__bhsp_2]),discharge_asm(discharge,[dt_c2_10__bhsp_2])],[dt_c2_10__bhsp_2,i2_10__bhsp_2]), [interesting(0.8),i1_10__bhsp_2]). fof(i1_10__bhsp_2,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ) => ( v1_bhsp_2(A,c1_10__bhsp_2) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(c1_10__bhsp_2)) & ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,A,E),B))) ) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_10__bhsp_2])],[i2_10_tmp__bhsp_2,dh_c2_10__bhsp_2]), [interesting(0.8),file(bhsp_2,i1_10__bhsp_2),[file(bhsp_2,i1_10__bhsp_2)]]). fof(i1_10_tmp__bhsp_2,plain, ( ( ~ v3_struct_0(c1_10__bhsp_2) & v3_rlvect_1(c1_10__bhsp_2) & v4_rlvect_1(c1_10__bhsp_2) & v5_rlvect_1(c1_10__bhsp_2) & v6_rlvect_1(c1_10__bhsp_2) & v7_rlvect_1(c1_10__bhsp_2) & v2_bhsp_1(c1_10__bhsp_2) & l1_bhsp_1(c1_10__bhsp_2) ) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_10__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_10__bhsp_2)) ) => ( v1_bhsp_2(A,c1_10__bhsp_2) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(c1_10__bhsp_2)) & ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( ~ 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,k3_bhsp_1(c1_10__bhsp_2,k6_rlvect_1(c1_10__bhsp_2,k2_normsp_1(c1_10__bhsp_2,A,E),B))) ) ) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_10__bhsp_2])],[dt_c1_10__bhsp_2,i1_10__bhsp_2]), [interesting(1),t9_bhsp_2]). fof(t9_bhsp_2,theorem,( ! [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] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m2_relset_1(B,k5_numbers,u1_struct_0(A)) ) => ( v1_bhsp_2(B,A) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(A)) & ! [D] : ( m1_subset_1(D,k1_numbers) => ~ ( ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) & r1_xreal_0(E,F) & r1_xreal_0(D,k3_bhsp_1(A,k6_rlvect_1(A,k2_normsp_1(A,B,F),C))) ) ) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_10_tmp__bhsp_2,dh_c1_10__bhsp_2]), [interesting(1),file(bhsp_2,t9_bhsp_2),[file(bhsp_2,t9_bhsp_2)]]).