% Mizar ND problem: t5_bhsp_2,bhsp_2,168,49 fof(dh_c1_6__bhsp_2,definition, ( ( ( ~ v3_struct_0(c1_6__bhsp_2) & v3_rlvect_1(c1_6__bhsp_2) & v4_rlvect_1(c1_6__bhsp_2) & v5_rlvect_1(c1_6__bhsp_2) & v6_rlvect_1(c1_6__bhsp_2) & v7_rlvect_1(c1_6__bhsp_2) & v2_bhsp_1(c1_6__bhsp_2) & l1_bhsp_1(c1_6__bhsp_2) ) => ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(B,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(B,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,B,A),c1_6__bhsp_2) ) ) ) ) => ! [C] : ( ( ~ v3_struct_0(C) & v3_rlvect_1(C) & v4_rlvect_1(C) & v5_rlvect_1(C) & v6_rlvect_1(C) & v7_rlvect_1(C) & v2_bhsp_1(C) & l1_bhsp_1(C) ) => ! [D] : ( m1_subset_1(D,k1_numbers) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,k5_numbers,u1_struct_0(C)) & m2_relset_1(E,k5_numbers,u1_struct_0(C)) ) => ( v1_bhsp_2(E,C) => v1_bhsp_2(k6_normsp_1(C,E,D),C) ) ) ) ) ), introduced(definition,[new_symbol(c1_6__bhsp_2),file(bhsp_2,c1_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c1_6__bhsp_2)]). fof(dh_c2_6__bhsp_2,definition, ( ( m1_subset_1(c2_6__bhsp_2,k1_numbers) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(A,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,A,c2_6__bhsp_2),c1_6__bhsp_2) ) ) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(C,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(C,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,C,B),c1_6__bhsp_2) ) ) ) ), introduced(definition,[new_symbol(c2_6__bhsp_2),file(bhsp_2,c2_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c2_6__bhsp_2)]). fof(dh_c3_6__bhsp_2,definition, ( ( ( v1_funct_1(c3_6__bhsp_2) & v1_funct_2(c3_6__bhsp_2,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(c3_6__bhsp_2,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(c3_6__bhsp_2,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c1_6__bhsp_2) ) ) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(A,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,A,c2_6__bhsp_2),c1_6__bhsp_2) ) ) ), introduced(definition,[new_symbol(c3_6__bhsp_2),file(bhsp_2,c3_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c3_6__bhsp_2)]). fof(e1_6__bhsp_2,assumption,( v1_bhsp_2(c3_6__bhsp_2,c1_6__bhsp_2) ), introduced(assumption,[file(bhsp_2,e1_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,e1_6__bhsp_2)]). 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(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(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_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_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(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_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). 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(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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_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_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_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(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(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(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(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_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(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). 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_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_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(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_normsp_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) & l2_rlvect_1(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,k1_numbers) ) => ( v1_funct_1(k6_normsp_1(A,B,C)) & v1_funct_2(k6_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) & m2_relset_1(k6_normsp_1(A,B,C),k5_numbers,u1_struct_0(A)) ) ) ), file(normsp_1,k6_normsp_1), [interesting(0.9),axiom,file(normsp_1,k6_normsp_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_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_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(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_6__bhsp_2,assumption, ( ~ v3_struct_0(c1_6__bhsp_2) & v3_rlvect_1(c1_6__bhsp_2) & v4_rlvect_1(c1_6__bhsp_2) & v5_rlvect_1(c1_6__bhsp_2) & v6_rlvect_1(c1_6__bhsp_2) & v7_rlvect_1(c1_6__bhsp_2) & v2_bhsp_1(c1_6__bhsp_2) & l1_bhsp_1(c1_6__bhsp_2) ), introduced(assumption,[file(bhsp_2,c1_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c1_6__bhsp_2)]). fof(dt_c2_6__bhsp_2,assumption,( m1_subset_1(c2_6__bhsp_2,k1_numbers) ), introduced(assumption,[file(bhsp_2,c2_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c2_6__bhsp_2)]). fof(dt_c3_6__bhsp_2,assumption, ( v1_funct_1(c3_6__bhsp_2) & v1_funct_2(c3_6__bhsp_2,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(c3_6__bhsp_2,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ), introduced(assumption,[file(bhsp_2,c3_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c3_6__bhsp_2)]). 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_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_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(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(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(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_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(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(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(dt_k3_rlvect_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l2_rlvect_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,k1_numbers) ) => m1_subset_1(k3_rlvect_1(A,B,C),u1_struct_0(A)) ) ), file(rlvect_1,k3_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k3_rlvect_1)]). 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(dh_c4_6__bhsp_2,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,D),A)) ) ) ) ) ) => ( m1_subset_1(c4_6__bhsp_2,u1_struct_0(c1_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,G),c4_6__bhsp_2)) ) ) ) ) ) ), introduced(definition,[new_symbol(c4_6__bhsp_2),file(bhsp_2,c4_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c4_6__bhsp_2)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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_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(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(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_6__bhsp_2,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(c1_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,D),A)) ) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_6__bhsp_2,dt_c3_6__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e1_6__bhsp_2,d1_bhsp_2]), [interesting(0.8),file(bhsp_2,e2_6__bhsp_2),[file(bhsp_2,e2_6__bhsp_2)]]). fof(dt_c4_6__bhsp_2,plain,( m1_subset_1(c4_6__bhsp_2,u1_struct_0(c1_6__bhsp_2)) ), inference(consider,[status(thm),assumptions([dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_6__bhsp_2,dt_c3_6__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,dh_c4_6__bhsp_2,e2_6__bhsp_2]), [interesting(0.8),file(bhsp_2,c4_6__bhsp_2),[file(bhsp_2,c4_6__bhsp_2)]]). fof(de_c5_6__bhsp_2,definition,( c5_6__bhsp_2 = k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2) ), introduced(definition,[new_symbol(c5_6__bhsp_2),file(bhsp_2,c5_6__bhsp_2)]), [interesting(0.8),axiom,file(bhsp_2,c5_6__bhsp_2)]). fof(dt_c5_6__bhsp_2,plain,( m1_subset_1(c5_6__bhsp_2,u1_struct_0(c1_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[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,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_l2_struct_0,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc6_membered,rc1_membered,rc1_xreal_0,rc4_struct_0,t1_subset,existence_l1_rlvect_1,dt_l1_rlvect_1,cc10_membered,cc11_membered,cc15_membered,cc4_membered,t2_subset,t6_boole,t7_boole,t8_boole,existence_l1_bhsp_1,existence_l1_struct_0,existence_l2_rlvect_1,dt_k1_numbers,dt_l1_bhsp_1,dt_l1_struct_0,dt_l2_rlvect_1,fc1_struct_0,fc2_membered,rc3_struct_0,existence_m1_subset_1,dt_k3_rlvect_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c4_6__bhsp_2,de_c5_6__bhsp_2]), [interesting(0.8),file(bhsp_2,c5_6__bhsp_2),[file(bhsp_2,c5_6__bhsp_2)]]). fof(e1_6_2__bhsp_2,assumption,( c2_6__bhsp_2 != 0 ), introduced(assumption,[file(bhsp_2,e1_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e1_6_2__bhsp_2)]). fof(dh_c1_6_2__bhsp_2,definition, ( ( m1_subset_1(c1_6_2__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c1_6_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(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),E),c5_6__bhsp_2)) ) ) ) ) ), introduced(definition,[new_symbol(c1_6_2__bhsp_2),file(bhsp_2,c1_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c1_6_2__bhsp_2)]). fof(e3_6_2__bhsp_2,assumption,( ~ r1_xreal_0(c1_6_2__bhsp_2,0) ), introduced(assumption,[file(bhsp_2,e3_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e3_6_2__bhsp_2)]). fof(dt_c1_6_2__bhsp_2,assumption,( m1_subset_1(c1_6_2__bhsp_2,k1_numbers) ), introduced(assumption,[file(bhsp_2,c1_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c1_6_2__bhsp_2)]). fof(dh_c2_6_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(k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2)),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,B),c4_6__bhsp_2)) ) ) ) => ( m2_subset_1(c2_6_2__bhsp_2,k1_numbers,k5_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_6_2__bhsp_2,C) & r1_xreal_0(k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2)),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,C),c4_6__bhsp_2)) ) ) ) ), introduced(definition,[new_symbol(c2_6_2__bhsp_2),file(bhsp_2,c2_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c2_6_2__bhsp_2)]). 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(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(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_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k7_xcmplx_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(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(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(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(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(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_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_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(e5_6_2__bhsp_2,plain,( k6_real_1(0,k18_complex1(c2_6__bhsp_2)) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc30_xreal_0,fc6_xreal_0,rc1_arytm_3,rc1_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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,t1_subset,t3_subset,t4_subset,t5_subset,projectivity_k16_complex1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_k7_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,fc2_membered,t1_numerals,t2_subset,t5_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,redefinition_k18_complex1,redefinition_k6_real_1,dt_k18_complex1,dt_k6_real_1,dt_c2_6__bhsp_2,spc0_numerals,spc0_boole]), [interesting(0.65),file(bhsp_2,e5_6_2__bhsp_2),[file(bhsp_2,e5_6_2__bhsp_2)]]). 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(t133_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ( ~ ( A != 0 & r1_xreal_0(k17_complex1(A),0) ) & ~ ( ~ r1_xreal_0(k17_complex1(A),0) & A = 0 ) ) ) ), file(complex1,t133_complex1), [interesting(0.9),axiom,file(complex1,t133_complex1)]). fof(e2_6_2__bhsp_2,plain,( ~ r1_xreal_0(k18_complex1(c2_6__bhsp_2),0) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_2__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,rc1_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,projectivity_k17_complex1,projectivity_k18_complex1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k17_complex1,redefinition_k18_complex1,dt_k17_complex1,dt_k18_complex1,dt_c2_6__bhsp_2,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_numerals,spc0_boole,e1_6_2__bhsp_2,t133_complex1]), [interesting(0.65),file(bhsp_2,e2_6_2__bhsp_2),[file(bhsp_2,e2_6_2__bhsp_2)]]). 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_6_2__bhsp_2,plain,( ~ r1_xreal_0(k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2)),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc5_membered,fc6_membered,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc30_xreal_0,rc1_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t5_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k6_real_1,dt_k18_complex1,dt_k6_real_1,dt_k7_xcmplx_0,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,cc2_xreal_0,fc6_xreal_0,spc0_numerals,spc0_boole,e5_6_2__bhsp_2,e2_6_2__bhsp_2,e3_6_2__bhsp_2,t73_real_1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.65),file(bhsp_2,e6_6_2__bhsp_2),[file(bhsp_2,e6_6_2__bhsp_2)]]). fof(e3_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,C),c4_6__bhsp_2)) ) ) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,dh_c4_6__bhsp_2,e2_6__bhsp_2]), [interesting(0.8),file(bhsp_2,e3_6__bhsp_2),[file(bhsp_2,e3_6__bhsp_2)]]). fof(e7_6_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(k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2)),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,B),c4_6__bhsp_2)) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,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_m2_relset_1,redefinition_m2_relset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_k7_xcmplx_0,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,fc30_xreal_0,fc5_membered,fc6_xreal_0,rc1_xreal_0,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t5_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k6_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e6_6_2__bhsp_2,e3_6__bhsp_2]), [interesting(0.65),file(bhsp_2,e7_6_2__bhsp_2),[file(bhsp_2,e7_6_2__bhsp_2)]]). fof(dt_c2_6_2__bhsp_2,plain,( m2_subset_1(c2_6_2__bhsp_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dh_c2_6_2__bhsp_2,e7_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,c2_6_2__bhsp_2),[file(bhsp_2,c2_6_2__bhsp_2)]]). fof(de_c3_6_2__bhsp_2,definition,( c3_6_2__bhsp_2 = c2_6_2__bhsp_2 ), introduced(definition,[new_symbol(c3_6_2__bhsp_2),file(bhsp_2,c3_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c3_6_2__bhsp_2)]). fof(dt_c3_6_2__bhsp_2,plain,( m2_subset_1(c3_6_2__bhsp_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_c2_6_2__bhsp_2,fc2_membered,de_c3_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,c3_6_2__bhsp_2),[file(bhsp_2,c3_6_2__bhsp_2)]]). fof(dh_c4_6_2__bhsp_2,definition, ( ( m2_subset_1(c4_6_2__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_2__bhsp_2,c4_6_2__bhsp_2) & r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_2__bhsp_2),c5_6__bhsp_2)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_2__bhsp_2,A) & r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),A),c5_6__bhsp_2)) ) ) ), introduced(definition,[new_symbol(c4_6_2__bhsp_2),file(bhsp_2,c4_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c4_6_2__bhsp_2)]). fof(e9_6_2__bhsp_2,assumption,( r1_xreal_0(c3_6_2__bhsp_2,c4_6_2__bhsp_2) ), introduced(assumption,[file(bhsp_2,e9_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e9_6_2__bhsp_2)]). fof(dt_c4_6_2__bhsp_2,assumption,( m2_subset_1(c4_6_2__bhsp_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_2,c4_6_2__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c4_6_2__bhsp_2)]). 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(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(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(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(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__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_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__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(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(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(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(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(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(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(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(commutativity_k4_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_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_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_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_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_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_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_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(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(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_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__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_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__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_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(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(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_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__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__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(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__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(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(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(fc26_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & ~ v3_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc26_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc26_xreal_0)]). fof(fc25_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & ~ v2_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc25_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc25_xreal_0)]). fof(fc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc2_xreal_0)]). fof(involutiveness_k2_real_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => k2_real_1(k2_real_1(A)) = A ) ), file(real_1,k2_real_1), [interesting(0.9),axiom,file(real_1,k2_real_1)]). fof(involutiveness_k5_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k5_xcmplx_0(k5_xcmplx_0(A)) = A ) ), file(xcmplx_0,k5_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k5_xcmplx_0)]). fof(redefinition_k2_real_1,definition,( ! [A] : ( m1_subset_1(A,k1_numbers) => k2_real_1(A) = k5_xcmplx_0(A) ) ), file(real_1,k2_real_1), [interesting(0.9),axiom,file(real_1,k2_real_1)]). fof(dt_k2_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k2_real_1(A),k1_numbers) ) ), file(real_1,k2_real_1), [interesting(0.9),axiom,file(real_1,k2_real_1)]). fof(dt_k5_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k5_xcmplx_0(A)) ) ), file(xcmplx_0,k5_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k5_xcmplx_0)]). fof(spc10_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(k5_xcmplx_0(A),k5_xcmplx_0(B)) = k5_xcmplx_0(k3_xcmplx_0(A,B)) ) ), file(arithm,spc10_arithm), [interesting(0.9),axiom,file(arithm,spc10_arithm)]). fof(spc11_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k7_xcmplx_0(k5_xcmplx_0(A),k5_xcmplx_0(B)) = k7_xcmplx_0(B,A) ) ), file(arithm,spc11_arithm), [interesting(0.9),axiom,file(arithm,spc11_arithm)]). fof(spc12_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,k5_xcmplx_0(B)) = k7_xcmplx_0(A,B) ) ), file(arithm,spc12_arithm), [interesting(0.9),axiom,file(arithm,spc12_arithm)]). fof(spc3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(1,A) = k5_xcmplx_0(A) ) ), file(arithm,spc3_arithm), [interesting(0.9),axiom,file(arithm,spc3_arithm)]). fof(d9_xcmplx_0,definition,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ), file(xcmplx_0,d9_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,d9_xcmplx_0)]). 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(e1_6_2_2__bhsp_2,plain,( k4_real_1(k18_complex1(c2_6__bhsp_2),k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2))) = k4_real_1(k18_complex1(c2_6__bhsp_2),k4_real_1(k2_real_1(k18_complex1(c2_6__bhsp_2)),c1_6_2__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc25_xreal_0,fc2_xreal_0,fc30_xreal_0,fc4_xreal_0,fc6_xreal_0,rc1_arytm_3,rc1_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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,t1_subset,t3_subset,t4_subset,t5_subset,projectivity_k16_complex1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,fc2_membered,t2_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k5_xcmplx_0,redefinition_k18_complex1,redefinition_k2_real_1,redefinition_k4_real_1,redefinition_k6_real_1,dt_k18_complex1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_xcmplx_0,dt_k6_real_1,dt_k7_xcmplx_0,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc10_arithm,spc11_arithm,spc12_arithm,spc3_arithm,spc4_arithm,spc7_arithm,t3_arithm,t6_arithm,spc1_numerals,spc1_boole,d9_xcmplx_0,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_2,e1_6_2_2__bhsp_2),[file(bhsp_2,e1_6_2_2__bhsp_2)]]). fof(e2_6_2_2__bhsp_2,plain,( k4_real_1(k18_complex1(c2_6__bhsp_2),k4_real_1(k2_real_1(k18_complex1(c2_6__bhsp_2)),c1_6_2__bhsp_2)) = k4_real_1(k4_real_1(k18_complex1(c2_6__bhsp_2),k2_real_1(k18_complex1(c2_6__bhsp_2))),c1_6_2__bhsp_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc25_xreal_0,fc2_xreal_0,fc4_xreal_0,rc1_arytm_3,rc1_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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,t1_subset,t3_subset,t4_subset,t5_subset,projectivity_k16_complex1,involutiveness_k5_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_k5_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,fc2_membered,spc10_arithm,spc7_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,redefinition_k18_complex1,redefinition_k2_real_1,redefinition_k4_real_1,dt_k18_complex1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,spc1_numerals,spc1_boole,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_2,e2_6_2_2__bhsp_2),[file(bhsp_2,e2_6_2_2__bhsp_2)]]). fof(e4_6_2__bhsp_2,plain,( k18_complex1(c2_6__bhsp_2) != 0 ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_2__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,rc1_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,projectivity_k17_complex1,projectivity_k18_complex1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k17_complex1,redefinition_k18_complex1,dt_k17_complex1,dt_k18_complex1,dt_c2_6__bhsp_2,spc0_numerals,spc0_boole,e1_6_2__bhsp_2,t133_complex1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.65),file(bhsp_2,e4_6_2__bhsp_2),[file(bhsp_2,e4_6_2__bhsp_2)]]). fof(d7_xcmplx_0,definition,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( ( A != 0 => ( B = k5_xcmplx_0(A) <=> k3_xcmplx_0(A,B) = 1 ) ) & ( A = 0 => ( B = k5_xcmplx_0(A) <=> B = 0 ) ) ) ) ) ), file(xcmplx_0,d7_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,d7_xcmplx_0)]). fof(e3_6_2_2__bhsp_2,plain,( k4_real_1(k4_real_1(k18_complex1(c2_6__bhsp_2),k2_real_1(k18_complex1(c2_6__bhsp_2))),c1_6_2__bhsp_2) = k4_real_1(1,c1_6_2__bhsp_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc25_xreal_0,fc2_xreal_0,fc4_xreal_0,rc1_arytm_3,rc1_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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,t1_subset,t3_subset,t4_subset,t5_subset,projectivity_k16_complex1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,fc2_membered,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k5_xcmplx_0,redefinition_k18_complex1,redefinition_k2_real_1,redefinition_k4_real_1,dt_k18_complex1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_xcmplx_0,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc10_arithm,spc7_arithm,t2_arithm,t3_arithm,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e4_6_2__bhsp_2,d7_xcmplx_0,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_2,e3_6_2_2__bhsp_2),[file(bhsp_2,e3_6_2_2__bhsp_2)]]). fof(e4_6_2_2__bhsp_2,plain,( k4_real_1(1,c1_6_2__bhsp_2) = c1_6_2__bhsp_2 ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_2__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc4_xreal_0,rc1_arytm_3,rc1_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_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,t1_subset,t3_subset,t4_subset,t5_subset,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,fc2_membered,spc7_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,commutativity_k3_xcmplx_0,commutativity_k4_real_1,redefinition_k4_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_c1_6_2__bhsp_2,spc1_numerals,spc1_boole,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_2,e4_6_2_2__bhsp_2),[file(bhsp_2,e4_6_2_2__bhsp_2)]]). fof(e12_6_2__bhsp_2,plain,( k4_real_1(k18_complex1(c2_6__bhsp_2),k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2))) = c1_6_2__bhsp_2 ), inference(iterative_eq,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_2__bhsp_2,dt_c1_6_2__bhsp_2])],[e1_6_2_2__bhsp_2,e2_6_2_2__bhsp_2,e3_6_2_2__bhsp_2,e4_6_2_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e12_6_2__bhsp_2),[file(bhsp_2,e12_6_2__bhsp_2)]]). fof(e8_6_2__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_6_2__bhsp_2,A) & r1_xreal_0(k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2)),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,A),c4_6__bhsp_2)) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dh_c2_6_2__bhsp_2,e7_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e8_6_2__bhsp_2),[file(bhsp_2,e8_6_2__bhsp_2)]]). fof(e10_6_2__bhsp_2,plain,( ~ r1_xreal_0(k6_real_1(c1_6_2__bhsp_2,k18_complex1(c2_6__bhsp_2)),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c4_6_2__bhsp_2,e9_6_2__bhsp_2,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc30_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,projectivity_k16_complex1,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_k16_complex1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_k7_xcmplx_0,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,fc6_xreal_0,rc1_xreal_0,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k18_complex1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k6_real_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c2_6_2__bhsp_2,dt_c3_6__bhsp_2,dt_c3_6_2__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_2__bhsp_2,de_c3_6_2__bhsp_2,fc2_membered,e9_6_2__bhsp_2,e8_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,e10_6_2__bhsp_2),[file(bhsp_2,e10_6_2__bhsp_2)]]). 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_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(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(e1_6_2_1__bhsp_2,plain,( k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c2_6__bhsp_2),k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2)) = k3_bhsp_1(c1_6__bhsp_2,k6_rlvect_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c2_6__bhsp_2),k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_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_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,cc9_membered,fc1_ordinal2,fc5_membered,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,fc1_struct_0,fc2_membered,rc3_struct_0,commutativity_k5_bhsp_1,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_k3_rlvect_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_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_2__bhsp_2,d5_bhsp_1]), [interesting(0.5),file(bhsp_2,e1_6_2_1__bhsp_2),[file(bhsp_2,e1_6_2_1__bhsp_2)]]). fof(t48_rlvect_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & l2_rlvect_1(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k3_rlvect_1(B,k6_rlvect_1(B,C,D),A) = k6_rlvect_1(B,k3_rlvect_1(B,C,A),k3_rlvect_1(B,D,A)) ) ) ) ) ), file(rlvect_1,t48_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,t48_rlvect_1)]). fof(e2_6_2_1__bhsp_2,plain,( k3_bhsp_1(c1_6__bhsp_2,k6_rlvect_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c2_6__bhsp_2),k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2))) = k3_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k6_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2),c2_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,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_arytm_3,cc3_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc1_xreal_0,rc4_struct_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,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_k5_numbers,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc4_membered,fc1_struct_0,rc3_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_l2_rlvect_1,existence_m1_subset_1,redefinition_k2_normsp_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k3_rlvect_1,dt_k6_rlvect_1,dt_l2_rlvect_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_2__bhsp_2,fc2_membered,t48_rlvect_1]), [interesting(0.5),file(bhsp_2,e2_6_2_1__bhsp_2),[file(bhsp_2,e2_6_2_1__bhsp_2)]]). fof(t33_bhsp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & v2_bhsp_1(B) & l1_bhsp_1(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => k3_bhsp_1(B,k3_rlvect_1(B,C,A)) = k4_real_1(k18_complex1(A),k3_bhsp_1(B,C)) ) ) ) ), file(bhsp_1,t33_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,t33_bhsp_1)]). fof(e3_6_2_1__bhsp_2,plain,( k3_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k6_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2),c2_6__bhsp_2)) = k4_real_1(k18_complex1(c2_6__bhsp_2),k3_bhsp_1(c1_6__bhsp_2,k6_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,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_arytm_3,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc23_xreal_0,fc4_xreal_0,fc5_membered,fc6_membered,rc1_membered,rc1_xreal_0,rc4_struct_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,projectivity_k16_complex1,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_k16_complex1,dt_k1_funct_1,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,cc10_membered,cc11_membered,cc15_membered,cc4_membered,fc1_struct_0,rc3_struct_0,spc7_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k4_real_1,dt_k18_complex1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k3_rlvect_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k6_rlvect_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_2__bhsp_2,fc2_membered,spc1_numerals,spc1_boole,t33_bhsp_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_2,e3_6_2_1__bhsp_2),[file(bhsp_2,e3_6_2_1__bhsp_2)]]). fof(e4_6_2_1__bhsp_2,plain,( k4_real_1(k18_complex1(c2_6__bhsp_2),k3_bhsp_1(c1_6__bhsp_2,k6_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2))) = k4_real_1(k18_complex1(c2_6__bhsp_2),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc4_xreal_0,rc1_arytm_3,rc1_xreal_0,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l2_struct_0,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,cc3_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc4_struct_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,projectivity_k16_complex1,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_k16_complex1,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,cc15_membered,fc1_struct_0,fc2_membered,rc3_struct_0,spc7_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,commutativity_k5_bhsp_1,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k4_real_1,redefinition_k5_bhsp_1,dt_k18_complex1,dt_k2_normsp_1,dt_k3_bhsp_1,dt_k3_xcmplx_0,dt_k4_bhsp_1,dt_k4_real_1,dt_k5_bhsp_1,dt_k6_rlvect_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_2__bhsp_2,spc1_numerals,spc1_boole,d5_bhsp_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(bhsp_2,e4_6_2_1__bhsp_2),[file(bhsp_2,e4_6_2_1__bhsp_2)]]). fof(e11_6_2__bhsp_2,plain,( k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c2_6__bhsp_2),k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2)) = k4_real_1(k18_complex1(c2_6__bhsp_2),k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c4_6__bhsp_2)) ), inference(iterative_eq,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[e1_6_2_1__bhsp_2,e2_6_2_1__bhsp_2,e3_6_2_1__bhsp_2,e4_6_2_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e11_6_2__bhsp_2),[file(bhsp_2,e11_6_2__bhsp_2)]]). fof(t70_xreal_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(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ), file(xreal_1,t70_xreal_1), [interesting(0.9),axiom,file(xreal_1,t70_xreal_1)]). 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__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(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(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(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(e13_6_2__bhsp_2,plain,( ~ r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_2__bhsp_2),c2_6__bhsp_2),c5_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([e9_6_2__bhsp_2,dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[reflexivity_r1_tarski,existence_l2_struct_0,dt_l2_struct_0,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,rc4_struct_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_rlvect_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,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,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,fc6_membered,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,projectivity_k16_complex1,existence_l1_bhsp_1,existence_l1_struct_0,existence_l2_rlvect_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k16_complex1,dt_k1_funct_1,dt_k1_numbers,dt_k4_bhsp_1,dt_k5_numbers,dt_k7_xcmplx_0,dt_l1_bhsp_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc23_xreal_0,fc2_membered,fc30_xreal_0,fc6_xreal_0,rc1_xreal_0,rc3_struct_0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_arithm,t4_real,t5_arithm,t6_arithm,t6_boole,t7_boole,t8_boole,projectivity_k18_complex1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,commutativity_k5_bhsp_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k18_complex1,redefinition_k2_normsp_1,redefinition_k4_real_1,redefinition_k5_bhsp_1,redefinition_k6_real_1,dt_k18_complex1,dt_k2_normsp_1,dt_k3_rlvect_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_bhsp_1,dt_k6_real_1,dt_k6_xcmplx_0,dt_c1_6__bhsp_2,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_2__bhsp_2,dt_c5_6__bhsp_2,de_c5_6__bhsp_2,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,fc5_xreal_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,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__rm1_r0_rm1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e12_6_2__bhsp_2,e2_6_2__bhsp_2,e10_6_2__bhsp_2,e11_6_2__bhsp_2,t70_xreal_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0]), [interesting(0.65),file(bhsp_2,e13_6_2__bhsp_2),[file(bhsp_2,e13_6_2__bhsp_2)]]). fof(d8_normsp_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) & l2_rlvect_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)) ) => ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,u1_struct_0(A)) & m2_relset_1(D,k5_numbers,u1_struct_0(A)) ) => ( D = k6_normsp_1(A,B,C) <=> ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k2_normsp_1(A,D,E) = k3_rlvect_1(A,k2_normsp_1(A,B,E),C) ) ) ) ) ) ) ), file(normsp_1,d8_normsp_1), [interesting(0.9),axiom,file(normsp_1,d8_normsp_1)]). fof(e14_6_2__bhsp_2,plain,( ~ r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_2__bhsp_2),c5_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([e9_6_2__bhsp_2,dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_l2_struct_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,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,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_c4_6__bhsp_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,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_l2_rlvect_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_k3_rlvect_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k6_normsp_1,dt_l2_rlvect_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6_2__bhsp_2,dt_c5_6__bhsp_2,de_c5_6__bhsp_2,fc2_membered,e13_6_2__bhsp_2,d8_normsp_1]), [interesting(0.65),file(bhsp_2,e14_6_2__bhsp_2),[file(bhsp_2,e14_6_2__bhsp_2)]]). fof(i7_6_2__bhsp_2,theorem,( $true ), introduced(tautology,[file(bhsp_2,i7_6_2__bhsp_2)]), [interesting(0.65),trivial,file(bhsp_2,i7_6_2__bhsp_2)]). fof(i6_6_2__bhsp_2,plain,( ~ r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_2__bhsp_2),c5_6__bhsp_2)) ), inference(conclusion,[status(thm),assumptions([e9_6_2__bhsp_2,dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2])],[e14_6_2__bhsp_2,i7_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i6_6_2__bhsp_2),[file(bhsp_2,i6_6_2__bhsp_2)]]). fof(i5_6_2__bhsp_2,plain,( ~ ( r1_xreal_0(c3_6_2__bhsp_2,c4_6_2__bhsp_2) & r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_2__bhsp_2),c5_6__bhsp_2)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_2__bhsp_2]),discharge_asm(discharge,[e9_6_2__bhsp_2])],[e9_6_2__bhsp_2,i6_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i5_6_2__bhsp_2),[file(bhsp_2,i5_6_2__bhsp_2)]]). fof(i5_6_2_tmp__bhsp_2,plain, ( m2_subset_1(c4_6_2__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_2__bhsp_2,c4_6_2__bhsp_2) & r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_2__bhsp_2),c5_6__bhsp_2)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[dt_c4_6_2__bhsp_2])],[dt_c4_6_2__bhsp_2,i5_6_2__bhsp_2]), [interesting(0.65),i4_6_2__bhsp_2]). fof(i4_6_2__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_2__bhsp_2,A) & r1_xreal_0(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),A),c5_6__bhsp_2)) ) ) ), inference(let,[status(thm),assumptions([dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[i5_6_2_tmp__bhsp_2,dh_c4_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i4_6_2__bhsp_2),[file(bhsp_2,i4_6_2__bhsp_2)]]). fof(i3_6_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(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__bhsp_2)) ) ) ) ), inference(take,[status(thm),assumptions([dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,e3_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_k2_zfmisc_1,dt_l1_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_l2_rlvect_1,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_k6_normsp_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c3_6_2__bhsp_2,dt_c5_6__bhsp_2,fc2_membered,i4_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i3_6_2__bhsp_2),[file(bhsp_2,i3_6_2__bhsp_2)]]). fof(i2_6_2__bhsp_2,plain,( ~ ( ~ r1_xreal_0(c1_6_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(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__bhsp_2)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_6_2__bhsp_2,e1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[e3_6_2__bhsp_2])],[e3_6_2__bhsp_2,i3_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i2_6_2__bhsp_2),[file(bhsp_2,i2_6_2__bhsp_2)]]). fof(i2_6_2_tmp__bhsp_2,plain, ( m1_subset_1(c1_6_2__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c1_6_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(c1_6_2__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__bhsp_2)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([e1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[dt_c1_6_2__bhsp_2])],[dt_c1_6_2__bhsp_2,i2_6_2__bhsp_2]), [interesting(0.65),i1_6_2__bhsp_2]). fof(i1_6_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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(let,[status(thm),assumptions([e1_6_2__bhsp_2,dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[i2_6_2_tmp__bhsp_2,dh_c1_6_2__bhsp_2]), [interesting(0.65),file(bhsp_2,i1_6_2__bhsp_2),[file(bhsp_2,i1_6_2__bhsp_2)]]). fof(i1_6_2_tmp__bhsp_2,plain, ( c2_6__bhsp_2 != 0 => ! [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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[e1_6_2__bhsp_2])],[e1_6_2__bhsp_2,i1_6_2__bhsp_2]), [interesting(0.8),e5_6__bhsp_2]). fof(e5_6__bhsp_2,plain, ( c2_6__bhsp_2 != 0 => ! [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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[i1_6_2_tmp__bhsp_2,dt_l2_struct_0,rc4_struct_0,dt_k2_zfmisc_1,dt_l1_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_l2_rlvect_1,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,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_k6_normsp_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c5_6__bhsp_2,fc2_membered,spc0_numerals,spc0_boole]), [interesting(0.8),file(bhsp_2,e5_6__bhsp_2),[file(bhsp_2,e5_6__bhsp_2)]]). fof(e1_6_1__bhsp_2,assumption,( c2_6__bhsp_2 = 0 ), introduced(assumption,[file(bhsp_2,e1_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e1_6_1__bhsp_2)]). fof(dh_c1_6_1__bhsp_2,definition, ( ( m1_subset_1(c1_6_1__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c1_6_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(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),E),c5_6__bhsp_2)) ) ) ) ) ), introduced(definition,[new_symbol(c1_6_1__bhsp_2),file(bhsp_2,c1_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c1_6_1__bhsp_2)]). fof(e2_6_1__bhsp_2,assumption,( ~ r1_xreal_0(c1_6_1__bhsp_2,0) ), introduced(assumption,[file(bhsp_2,e2_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e2_6_1__bhsp_2)]). fof(dt_c1_6_1__bhsp_2,assumption,( m1_subset_1(c1_6_1__bhsp_2,k1_numbers) ), introduced(assumption,[file(bhsp_2,c1_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c1_6_1__bhsp_2)]). fof(dh_c2_6_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(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,B),c4_6__bhsp_2)) ) ) ) => ( m2_subset_1(c2_6_1__bhsp_2,k1_numbers,k5_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_6_1__bhsp_2,C) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,C),c4_6__bhsp_2)) ) ) ) ), introduced(definition,[new_symbol(c2_6_1__bhsp_2),file(bhsp_2,c2_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c2_6_1__bhsp_2)]). fof(e3_6_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(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,B),c4_6__bhsp_2)) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_6__bhsp_2,dt_c1_6_1__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e2_6_1__bhsp_2,e3_6__bhsp_2]), [interesting(0.65),file(bhsp_2,e3_6_1__bhsp_2),[file(bhsp_2,e3_6_1__bhsp_2)]]). fof(dt_c2_6_1__bhsp_2,plain,( m2_subset_1(c2_6_1__bhsp_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dh_c2_6_1__bhsp_2,e3_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,c2_6_1__bhsp_2),[file(bhsp_2,c2_6_1__bhsp_2)]]). fof(de_c3_6_1__bhsp_2,definition,( c3_6_1__bhsp_2 = c2_6_1__bhsp_2 ), introduced(definition,[new_symbol(c3_6_1__bhsp_2),file(bhsp_2,c3_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c3_6_1__bhsp_2)]). fof(dt_c3_6_1__bhsp_2,plain,( m2_subset_1(c3_6_1__bhsp_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_c2_6_1__bhsp_2,fc2_membered,de_c3_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,c3_6_1__bhsp_2),[file(bhsp_2,c3_6_1__bhsp_2)]]). fof(dh_c4_6_1__bhsp_2,definition, ( ( m2_subset_1(c4_6_1__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_1__bhsp_2,c4_6_1__bhsp_2) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_1__bhsp_2),c5_6__bhsp_2)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_1__bhsp_2,A) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),A),c5_6__bhsp_2)) ) ) ), introduced(definition,[new_symbol(c4_6_1__bhsp_2),file(bhsp_2,c4_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c4_6_1__bhsp_2)]). fof(e5_6_1__bhsp_2,assumption,( r1_xreal_0(c3_6_1__bhsp_2,c4_6_1__bhsp_2) ), introduced(assumption,[file(bhsp_2,e5_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,e5_6_1__bhsp_2)]). fof(dt_c4_6_1__bhsp_2,assumption,( m2_subset_1(c4_6_1__bhsp_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(bhsp_2,c4_6_1__bhsp_2)]), [interesting(0.65),axiom,file(bhsp_2,c4_6_1__bhsp_2)]). fof(dt_k1_rlvect_1,axiom,( ! [A] : ( l2_struct_0(A) => m1_subset_1(k1_rlvect_1(A),u1_struct_0(A)) ) ), file(rlvect_1,k1_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,k1_rlvect_1)]). fof(t23_rlvect_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( ~ v3_struct_0(B) & v3_rlvect_1(B) & v4_rlvect_1(B) & v5_rlvect_1(B) & v6_rlvect_1(B) & v7_rlvect_1(B) & l2_rlvect_1(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ( ( A = 0 | C = k1_rlvect_1(B) ) => k3_rlvect_1(B,C,A) = k1_rlvect_1(B) ) ) ) ) ), file(rlvect_1,t23_rlvect_1), [interesting(0.9),axiom,file(rlvect_1,t23_rlvect_1)]). fof(e1_6_1_1__bhsp_2,plain,( k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_1__bhsp_2),c2_6__bhsp_2),k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2)) = k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_1__bhsp_2),0),k1_rlvect_1(c1_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2,dt_c4_6_1__bhsp_2,e1_6_1__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,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_arytm_3,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc1_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_struct_0,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_k4_bhsp_1,dt_k5_numbers,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc4_membered,fc1_struct_0,rc3_struct_0,rc4_struct_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,existence_l2_rlvect_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,dt_k1_numbers,dt_k1_rlvect_1,dt_k2_normsp_1,dt_k3_rlvect_1,dt_k5_bhsp_1,dt_l2_rlvect_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_1__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,e1_6_1__bhsp_2,t23_rlvect_1]), [interesting(0.5),file(bhsp_2,e1_6_1_1__bhsp_2),[file(bhsp_2,e1_6_1_1__bhsp_2)]]). fof(e2_6_1_1__bhsp_2,plain,( k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_1__bhsp_2),0),k1_rlvect_1(c1_6__bhsp_2)) = k5_bhsp_1(c1_6__bhsp_2,k1_rlvect_1(c1_6__bhsp_2),k1_rlvect_1(c1_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6_1__bhsp_2])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,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_arytm_3,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc1_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,existence_l1_bhsp_1,existence_l1_rlvect_1,existence_l1_struct_0,existence_l2_struct_0,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_k4_bhsp_1,dt_k5_numbers,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_l2_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc4_membered,fc1_struct_0,rc3_struct_0,rc4_struct_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,existence_l2_rlvect_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,dt_k1_numbers,dt_k1_rlvect_1,dt_k2_normsp_1,dt_k3_rlvect_1,dt_k5_bhsp_1,dt_l2_rlvect_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6_1__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,t23_rlvect_1]), [interesting(0.5),file(bhsp_2,e2_6_1_1__bhsp_2),[file(bhsp_2,e2_6_1_1__bhsp_2)]]). fof(t41_bhsp_1,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] : ( m1_subset_1(B,u1_struct_0(A)) => k5_bhsp_1(A,B,B) = 0 ) ) ), file(bhsp_1,t41_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,t41_bhsp_1)]). fof(e3_6_1_1__bhsp_2,plain,( k5_bhsp_1(c1_6__bhsp_2,k1_rlvect_1(c1_6__bhsp_2),k1_rlvect_1(c1_6__bhsp_2)) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_6__bhsp_2])],[cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,rc1_arytm_3,rc1_xreal_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,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_arytm_3,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,existence_l1_struct_0,existence_l2_rlvect_1,existence_l2_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k4_bhsp_1,dt_k5_numbers,dt_l1_struct_0,dt_l2_rlvect_1,dt_l2_struct_0,dt_m2_subset_1,cc15_membered,fc1_struct_0,fc2_membered,rc3_struct_0,rc4_struct_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k5_bhsp_1,existence_l1_bhsp_1,existence_m1_subset_1,redefinition_k5_bhsp_1,dt_k1_rlvect_1,dt_k5_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,spc0_numerals,spc0_boole,t41_bhsp_1]), [interesting(0.5),file(bhsp_2,e3_6_1_1__bhsp_2),[file(bhsp_2,e3_6_1_1__bhsp_2)]]). fof(e7_6_1__bhsp_2,plain,( k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_1__bhsp_2),c2_6__bhsp_2),k3_rlvect_1(c1_6__bhsp_2,c4_6__bhsp_2,c2_6__bhsp_2)) = 0 ), inference(iterative_eq,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6__bhsp_2,e1_6_1__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6_1__bhsp_2,dt_c1_6__bhsp_2])],[e1_6_1_1__bhsp_2,e2_6_1_1__bhsp_2,e3_6_1_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e7_6_1__bhsp_2),[file(bhsp_2,e7_6_1__bhsp_2)]]). fof(e4_6_1__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_6_1__bhsp_2,A) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,A),c4_6__bhsp_2)) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dh_c2_6_1__bhsp_2,e3_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e4_6_1__bhsp_2),[file(bhsp_2,e4_6_1__bhsp_2)]]). fof(e6_6_1__bhsp_2,plain,( ~ r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_1__bhsp_2),c4_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c4_6_1__bhsp_2,e5_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_6__bhsp_2,dt_c1_6_1__bhsp_2,dt_c2_6_1__bhsp_2,dt_c3_6__bhsp_2,dt_c3_6_1__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_1__bhsp_2,de_c3_6_1__bhsp_2,fc2_membered,e5_6_1__bhsp_2,e4_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,e6_6_1__bhsp_2),[file(bhsp_2,e6_6_1__bhsp_2)]]). fof(t44_bhsp_1,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] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => r1_xreal_0(0,k5_bhsp_1(A,B,C)) ) ) ) ), file(bhsp_1,t44_bhsp_1), [interesting(0.9),axiom,file(bhsp_1,t44_bhsp_1)]). fof(e8_6_1__bhsp_2,plain,( ~ r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k3_rlvect_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c4_6_1__bhsp_2),c2_6__bhsp_2),c5_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c4_6_1__bhsp_2,e5_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[reflexivity_r1_tarski,existence_l2_struct_0,dt_l2_struct_0,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,rc4_struct_0,antisymmetry_r2_hidden,existence_l1_rlvect_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_rlvect_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,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k4_bhsp_1,dt_k5_numbers,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_struct_0,fc2_membered,rc3_struct_0,t1_numerals,t1_real,t2_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,redefinition_k2_normsp_1,redefinition_k5_bhsp_1,dt_k2_normsp_1,dt_k3_rlvect_1,dt_k5_bhsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c1_6_1__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6__bhsp_2,dt_c4_6_1__bhsp_2,dt_c5_6__bhsp_2,de_c5_6__bhsp_2,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_numerals,spc0_boole,e7_6_1__bhsp_2,e6_6_1__bhsp_2,t44_bhsp_1]), [interesting(0.65),file(bhsp_2,e8_6_1__bhsp_2),[file(bhsp_2,e8_6_1__bhsp_2)]]). fof(e9_6_1__bhsp_2,plain,( ~ r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_1__bhsp_2),c5_6__bhsp_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c4_6_1__bhsp_2,e5_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_l2_struct_0,dt_k1_xboole_0,dt_l2_struct_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,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,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_rlvect_1,dt_l1_struct_0,dt_m1_relset_1,dt_c4_6__bhsp_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,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_l2_rlvect_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_k3_rlvect_1,dt_k5_bhsp_1,dt_k5_numbers,dt_k6_normsp_1,dt_l2_rlvect_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c1_6_1__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c4_6_1__bhsp_2,dt_c5_6__bhsp_2,de_c5_6__bhsp_2,fc2_membered,e8_6_1__bhsp_2,d8_normsp_1]), [interesting(0.65),file(bhsp_2,e9_6_1__bhsp_2),[file(bhsp_2,e9_6_1__bhsp_2)]]). fof(i7_6_1__bhsp_2,theorem,( $true ), introduced(tautology,[file(bhsp_2,i7_6_1__bhsp_2)]), [interesting(0.65),trivial,file(bhsp_2,i7_6_1__bhsp_2)]). fof(i6_6_1__bhsp_2,plain,( ~ r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_1__bhsp_2),c5_6__bhsp_2)) ), inference(conclusion,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c4_6_1__bhsp_2,e5_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[e9_6_1__bhsp_2,i7_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i6_6_1__bhsp_2),[file(bhsp_2,i6_6_1__bhsp_2)]]). fof(i5_6_1__bhsp_2,plain,( ~ ( r1_xreal_0(c3_6_1__bhsp_2,c4_6_1__bhsp_2) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_1__bhsp_2),c5_6__bhsp_2)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c4_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[e5_6_1__bhsp_2])],[e5_6_1__bhsp_2,i6_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i5_6_1__bhsp_2),[file(bhsp_2,i5_6_1__bhsp_2)]]). fof(i5_6_1_tmp__bhsp_2,plain, ( m2_subset_1(c4_6_1__bhsp_2,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_1__bhsp_2,c4_6_1__bhsp_2) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c4_6_1__bhsp_2),c5_6__bhsp_2)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[dt_c4_6_1__bhsp_2])],[dt_c4_6_1__bhsp_2,i5_6_1__bhsp_2]), [interesting(0.65),i4_6_1__bhsp_2]). fof(i4_6_1__bhsp_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c3_6_1__bhsp_2,A) & r1_xreal_0(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),A),c5_6__bhsp_2)) ) ) ), inference(let,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[i5_6_1_tmp__bhsp_2,dh_c4_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i4_6_1__bhsp_2),[file(bhsp_2,i4_6_1__bhsp_2)]]). fof(i3_6_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(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__bhsp_2)) ) ) ) ), inference(take,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c1_6_1__bhsp_2,e2_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_k2_zfmisc_1,dt_l1_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_l2_rlvect_1,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_k6_normsp_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c1_6_1__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c3_6_1__bhsp_2,dt_c5_6__bhsp_2,fc2_membered,i4_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i3_6_1__bhsp_2),[file(bhsp_2,i3_6_1__bhsp_2)]]). fof(i2_6_1__bhsp_2,plain,( ~ ( ~ r1_xreal_0(c1_6_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(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__bhsp_2)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c1_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[e2_6_1__bhsp_2])],[e2_6_1__bhsp_2,i3_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i2_6_1__bhsp_2),[file(bhsp_2,i2_6_1__bhsp_2)]]). fof(i2_6_1_tmp__bhsp_2,plain, ( m1_subset_1(c1_6_1__bhsp_2,k1_numbers) => ~ ( ~ r1_xreal_0(c1_6_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(c1_6_1__bhsp_2,k5_bhsp_1(c1_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),B),c5_6__bhsp_2)) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[dt_c1_6_1__bhsp_2])],[dt_c1_6_1__bhsp_2,i2_6_1__bhsp_2]), [interesting(0.65),i1_6_1__bhsp_2]). fof(i1_6_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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c2_6__bhsp_2,e1_6_1__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[i2_6_1_tmp__bhsp_2,dh_c1_6_1__bhsp_2]), [interesting(0.65),file(bhsp_2,i1_6_1__bhsp_2),[file(bhsp_2,i1_6_1__bhsp_2)]]). fof(i1_6_1_tmp__bhsp_2,plain, ( c2_6__bhsp_2 = 0 => ! [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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2]),discharge_asm(discharge,[e1_6_1__bhsp_2])],[e1_6_1__bhsp_2,i1_6_1__bhsp_2]), [interesting(0.8),e4_6__bhsp_2]). fof(e4_6__bhsp_2,plain, ( c2_6__bhsp_2 = 0 => ! [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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[i1_6_1_tmp__bhsp_2,dt_l2_struct_0,rc4_struct_0,dt_k2_zfmisc_1,dt_l1_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_l2_rlvect_1,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,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_k6_normsp_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c5_6__bhsp_2,fc2_membered,spc0_numerals,spc0_boole]), [interesting(0.8),file(bhsp_2,e4_6__bhsp_2),[file(bhsp_2,e4_6__bhsp_2)]]). fof(e6_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_k2_zfmisc_1,dt_l1_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_l2_rlvect_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k3_rlvect_1,dt_k4_bhsp_1,dt_k5_ordinal2,dt_l1_bhsp_1,dt_l1_struct_0,dt_l2_rlvect_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,dt_c4_6__bhsp_2,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_k6_normsp_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c5_6__bhsp_2,de_c5_6__bhsp_2,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,e5_6__bhsp_2,e4_6__bhsp_2]), [interesting(0.8),file(bhsp_2,e6_6__bhsp_2),[file(bhsp_2,e6_6__bhsp_2)]]). fof(i6_6__bhsp_2,theorem,( $true ), introduced(tautology,[file(bhsp_2,i6_6__bhsp_2)]), [interesting(0.8),trivial,file(bhsp_2,i6_6__bhsp_2)]). fof(i5_6__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_6__bhsp_2,k2_normsp_1(c1_6__bhsp_2,k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),C),c5_6__bhsp_2)) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__bhsp_2])],[dt_l2_struct_0,rc4_struct_0,dt_k2_zfmisc_1,dt_l1_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_l2_rlvect_1,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,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_k6_normsp_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c5_6__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,e6_6__bhsp_2,i6_6__bhsp_2]), [interesting(0.8),file(bhsp_2,i5_6__bhsp_2),[file(bhsp_2,i5_6__bhsp_2)]]). fof(i4_6__bhsp_2,plain,( v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c1_6__bhsp_2) ), inference(take,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2,e1_6__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_k6_normsp_1,dt_l1_bhsp_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_6__bhsp_2,dt_c2_6__bhsp_2,dt_c3_6__bhsp_2,dt_c5_6__bhsp_2,fc2_membered,spc0_numerals,spc0_boole,d1_bhsp_2,i5_6__bhsp_2]), [interesting(0.8),file(bhsp_2,i4_6__bhsp_2),[file(bhsp_2,i4_6__bhsp_2)]]). fof(i3_6__bhsp_2,plain, ( v1_bhsp_2(c3_6__bhsp_2,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c1_6__bhsp_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2,dt_c3_6__bhsp_2]),discharge_asm(discharge,[e1_6__bhsp_2])],[e1_6__bhsp_2,i4_6__bhsp_2]), [interesting(0.8),file(bhsp_2,i3_6__bhsp_2),[file(bhsp_2,i3_6__bhsp_2)]]). fof(i3_6_tmp__bhsp_2,plain, ( ( v1_funct_1(c3_6__bhsp_2) & v1_funct_2(c3_6__bhsp_2,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(c3_6__bhsp_2,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(c3_6__bhsp_2,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,c3_6__bhsp_2,c2_6__bhsp_2),c1_6__bhsp_2) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2]),discharge_asm(discharge,[dt_c3_6__bhsp_2])],[dt_c3_6__bhsp_2,i3_6__bhsp_2]), [interesting(0.8),i2_6__bhsp_2]). fof(i2_6__bhsp_2,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(A,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,A,c2_6__bhsp_2),c1_6__bhsp_2) ) ) ), inference(let,[status(thm),assumptions([dt_c2_6__bhsp_2,dt_c1_6__bhsp_2])],[i3_6_tmp__bhsp_2,dh_c3_6__bhsp_2]), [interesting(0.8),file(bhsp_2,i2_6__bhsp_2),[file(bhsp_2,i2_6__bhsp_2)]]). fof(i2_6_tmp__bhsp_2,plain, ( m1_subset_1(c2_6__bhsp_2,k1_numbers) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(A,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(A,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,A,c2_6__bhsp_2),c1_6__bhsp_2) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_6__bhsp_2]),discharge_asm(discharge,[dt_c2_6__bhsp_2])],[dt_c2_6__bhsp_2,i2_6__bhsp_2]), [interesting(0.8),i1_6__bhsp_2]). fof(i1_6__bhsp_2,plain,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(B,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(B,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,B,A),c1_6__bhsp_2) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_6__bhsp_2])],[i2_6_tmp__bhsp_2,dh_c2_6__bhsp_2]), [interesting(0.8),file(bhsp_2,i1_6__bhsp_2),[file(bhsp_2,i1_6__bhsp_2)]]). fof(i1_6_tmp__bhsp_2,plain, ( ( ~ v3_struct_0(c1_6__bhsp_2) & v3_rlvect_1(c1_6__bhsp_2) & v4_rlvect_1(c1_6__bhsp_2) & v5_rlvect_1(c1_6__bhsp_2) & v6_rlvect_1(c1_6__bhsp_2) & v7_rlvect_1(c1_6__bhsp_2) & v2_bhsp_1(c1_6__bhsp_2) & l1_bhsp_1(c1_6__bhsp_2) ) => ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(c1_6__bhsp_2)) & m2_relset_1(B,k5_numbers,u1_struct_0(c1_6__bhsp_2)) ) => ( v1_bhsp_2(B,c1_6__bhsp_2) => v1_bhsp_2(k6_normsp_1(c1_6__bhsp_2,B,A),c1_6__bhsp_2) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_6__bhsp_2])],[dt_c1_6__bhsp_2,i1_6__bhsp_2]), [interesting(1),t5_bhsp_2]). fof(t5_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] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,u1_struct_0(A)) & m2_relset_1(C,k5_numbers,u1_struct_0(A)) ) => ( v1_bhsp_2(C,A) => v1_bhsp_2(k6_normsp_1(A,C,B),A) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_6_tmp__bhsp_2,dh_c1_6__bhsp_2]), [interesting(1),file(bhsp_2,t5_bhsp_2),[file(bhsp_2,t5_bhsp_2)]]).