% Mizar problem: t15_topreal1,topreal1,560,25 fof(t15_topreal1,conjecture,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ( B != C => r1_topreal1(k15_euclid(A),B,C,k3_topreal1(A,B,C)) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_metric_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_nat_1,cc1_pre_topc,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_int_1,cc2_membered,cc2_nat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k17_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_topreal1,commutativity_k3_xboole_0,commutativity_k3_xcmplx_0,commutativity_k4_metric_1,commutativity_k4_real_1,connectedness_r1_xreal_0,d10_pre_topc,d10_xboole_0,d11_euclid,d12_funct_1,d13_euclid,d1_connsp_2,d1_euclid,d1_funct_2,d1_metric_1,d1_rcomp_1,d1_topmetr,d1_xreal_0,d2_borsuk_1,d2_topmetr,d2_topreal1,d3_tarski,d3_topreal1,d4_euclid,d5_euclid,d5_funct_1,d6_euclid,d7_euclid,d8_euclid,d8_funct_1,d8_topmetr,d9_euclid,d9_pre_topc,dt_g1_metric_1,dt_g1_pre_topc,dt_k11_rvsum_1,dt_k12_euclid,dt_k13_euclid,dt_k14_euclid,dt_k15_euclid,dt_k15_rvsum_1,dt_k16_complex1,dt_k16_euclid,dt_k17_complex1,dt_k17_euclid,dt_k18_complex1,dt_k18_euclid,dt_k19_euclid,dt_k1_binop_1,dt_k1_euclid,dt_k1_funct_1,dt_k1_metric_1,dt_k1_numbers,dt_k1_rcomp_1,dt_k1_real_1,dt_k1_relat_1,dt_k1_topmetr,dt_k1_topreal1,dt_k1_tops_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_euclid,dt_k22_borsuk_1,dt_k2_finseq_2,dt_k2_metric_1,dt_k2_pre_topc,dt_k2_rcomp_1,dt_k2_relat_1,dt_k2_topmetr,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_pre_topc,dt_k3_real_1,dt_k3_topmetr,dt_k3_topreal1,dt_k3_xboole_0,dt_k3_xcmplx_0,dt_k4_euclid,dt_k4_finseq_2,dt_k4_finseqop,dt_k4_metric_1,dt_k4_pre_topc,dt_k4_real_1,dt_k4_relset_1,dt_k4_topmetr,dt_k4_xcmplx_0,dt_k5_euclid,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_real_1,dt_k5_topmetr,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_rvsum_1,dt_k7_xcmplx_0,dt_k8_euclid,dt_k8_funct_2,dt_k8_metric_1,dt_k8_square_1,dt_k9_euclid,dt_k9_metric_1,dt_k9_relat_1,dt_k9_rvsum_1,dt_k9_square_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_connsp_2,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,dt_m1_topmetr,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_connsp_2,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,existence_m1_topmetr,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_topmetr,fc1_topreal1,fc1_xboole_0,fc1_xreal_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc27_membered,fc27_xreal_0,fc28_membered,fc28_xreal_0,fc29_membered,fc29_xreal_0,fc2_euclid,fc2_int_1,fc2_membered,fc2_nat_1,fc2_pre_topc,fc2_topmetr,fc30_membered,fc30_xreal_0,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_int_1,fc3_metric_1,fc3_nat_1,fc3_pcomps_1,fc3_pre_topc,fc3_topmetr,fc3_xreal_0,fc4_int_1,fc4_metric_1,fc4_nat_1,fc4_pcomps_1,fc4_pre_topc,fc4_subset_1,fc4_xreal_0,fc5_borsuk_1,fc5_int_1,fc5_membered,fc5_pre_topc,fc5_xreal_0,fc6_int_1,fc6_membered,fc6_xreal_0,fc7_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,fraenkel_a_0_4_topreal1,fraenkel_a_2_0_rcomp_1,fraenkel_a_3_0_topreal1,free_g1_metric_1,free_g1_pre_topc,idempotence_k3_xboole_0,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,l1_topreal1,projectivity_k16_complex1,projectivity_k17_complex1,projectivity_k18_complex1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_topmetr,rc1_xboole_0,rc1_xreal_0,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_topmetr,rc2_xboole_0,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_pre_topc,rc3_struct_0,rc3_xreal_0,rc4_pre_topc,rc4_xreal_0,rc5_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,redefinition_k17_complex1,redefinition_k18_complex1,redefinition_k1_metric_1,redefinition_k1_real_1,redefinition_k3_real_1,redefinition_k3_topreal1,redefinition_k4_finseqop,redefinition_k4_metric_1,redefinition_k4_pre_topc,redefinition_k4_real_1,redefinition_k4_relset_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k5_topmetr,redefinition_k6_real_1,redefinition_k8_euclid,redefinition_k8_funct_2,redefinition_k9_euclid,redefinition_k9_square_1,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_zfrefle1__e5_19__topreal1,spc0_boole,spc1_boole,spc1_numerals,t11_heine,t11_relset_1,t127_real_2,t12_metric_1,t12_pre_topc,t12_relset_1,t12_topmetr,t138_complex1,t14_euclid,t14_topmetr,t15_rcomp_1,t15_topmetr,t16_tbsp_1,t1_arithm,t1_numerals,t1_real,t1_subset,t1_xboole_1,t20_topmetr,t21_topmetr,t22_topmetr,t26_compts_1,t27_topmetr,t2_arithm,t2_boole,t2_metric_1,t2_real,t2_subset,t2_tarski,t30_euclid,t31_euclid,t33_euclid,t37_euclid,t38_euclid,t38_pcomps_1,t3_arithm,t3_real,t3_subset,t3_topmetr,t44_euclid,t45_euclid,t49_euclid,t4_arithm,t4_real,t4_subset,t50_euclid,t53_euclid,t54_euclid,t54_tops_1,t54_xcmplx_1,t55_tops_1,t5_arithm,t5_metric_1,t5_real,t5_subset,t6_arithm,t6_boole,t6_real,t70_xreal_1,t7_boole,t7_real,t83_borsuk_1,t88_xcmplx_1,t8_boole,t8_real]), [file(topreal1,t15_topreal1)]). fof(abstractness_v1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), []). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc12_membered,axiom,( ! [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), []). fof(cc13_membered,axiom,( ! [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), []). fof(cc14_membered,axiom,( ! [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), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [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), []). fof(cc19_membered,axiom,( ! [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), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), []). fof(cc1_relset_1,axiom,( ! [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), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc20_membered,axiom,( ! [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), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_xreal_0,axiom,( ! [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), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_xreal_0,axiom,( ! [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), []). fof(cc5_xreal_0,axiom,( ! [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), []). fof(cc6_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). fof(cc6_xreal_0,axiom,( ! [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), []). fof(cc7_xreal_0,axiom,( ! [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), []). fof(cc8_xreal_0,axiom,( ! [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), []). fof(cc9_membered,axiom,( ! [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), []). fof(commutativity_k17_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k17_euclid(A,B,C) = k17_euclid(A,C,B) ) ), file(euclid,k17_euclid), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(commutativity_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), []). fof(commutativity_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k3_topreal1(A,C,B) ) ), file(topreal1,k3_topreal1), []). fof(commutativity_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), []). fof(commutativity_k3_xcmplx_0,axiom,( ! [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), []). fof(commutativity_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k4_metric_1(A,C,B) ) ), file(metric_1,k4_metric_1), []). fof(commutativity_k4_real_1,axiom,( ! [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), []). fof(connectedness_r1_xreal_0,axiom,( ! [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), []). fof(d10_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( ( v1_pre_topc(C) & m1_pre_topc(C,A) ) => ( C = k3_pre_topc(A,B) <=> k2_pre_topc(C) = B ) ) ) ) ), file(pre_topc,d10_pre_topc), []). fof(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). fof(d11_euclid,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(B))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(B))) => ( D = k18_euclid(A,B,C) <=> ! [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(B)) => ( E = C => D = k9_euclid(B,A,E) ) ) ) ) ) ) ) ), file(euclid,d11_euclid), []). fof(d12_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k9_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( r2_hidden(E,k1_relat_1(A)) & r2_hidden(E,B) & D = k1_funct_1(A,E) ) ) ) ) ), file(funct_1,d12_funct_1), []). fof(d13_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( D = k20_euclid(A,B,C) <=> ! [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(A)) => ! [F] : ( m2_finseq_2(F,k1_numbers,k1_euclid(A)) => ( ( E = B & F = C ) => D = k8_euclid(A,E,F) ) ) ) ) ) ) ) ) ), file(euclid,d13_euclid), []). fof(d1_connsp_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( m1_connsp_2(C,A,B) <=> r2_hidden(B,k1_tops_1(A,C)) ) ) ) ) ), file(connsp_2,d1_connsp_2), []). fof(d1_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ), file(euclid,d1_euclid), []). fof(d1_funct_2,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), []). fof(d1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k2_metric_1(A,B,C) = k1_metric_1(u1_struct_0(A),u1_struct_0(A),u1_metric_1(A),B,C) ) ) ) ), file(metric_1,d1_metric_1), []). fof(d1_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k1_rcomp_1(A,B) = a_2_0_rcomp_1(A,B) ) ) ), file(rcomp_1,d1_rcomp_1), []). fof(d1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) & l1_metric_1(B) ) => ( m1_topmetr(B,A) <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A)) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),C,D) = k1_binop_1(u1_metric_1(A),C,D) ) ) ) ) ) ) ), file(topmetr,d1_topmetr), []). fof(d1_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) <=> r2_hidden(A,k1_numbers) ) ), file(xreal_0,d1_xreal_0), []). fof(d2_borsuk_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v5_pre_topc(C,A,B) <=> ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_connsp_2(E,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) => ? [F] : ( m1_connsp_2(F,A,D) & r1_tarski(k4_pre_topc(A,B,C,F),E) ) ) ) ) ) ) ) ), file(borsuk_1,d2_borsuk_1), []). fof(d2_topmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( ( v1_metric_1(C) & m1_topmetr(C,A) ) => ( C = k1_topmetr(A,B) <=> u1_struct_0(C) = B ) ) ) ) ), file(topmetr,d2_topmetr), []). fof(d2_topreal1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( r1_topreal1(A,B,C,D) <=> ? [E] : ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D))) & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D))) & v3_tops_2(E,k5_topmetr,k3_pre_topc(A,D)) & k1_funct_1(E,0) = B & k1_funct_1(E,1) = C ) ) ) ) ) ) ), file(topreal1,d2_topreal1), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d3_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k1_topreal1(A,B,C) = a_3_0_topreal1(A,B,C) ) ) ) ), file(topreal1,d3_topreal1), []). fof(d4_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ), file(euclid,d4_euclid), []). fof(d5_euclid,axiom,( ! [A] : ( m2_finseq_1(A,k1_numbers) => k12_euclid(A) = k9_square_1(k15_rvsum_1(k11_rvsum_1(A))) ) ), file(euclid,d5_euclid), []). fof(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d6_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(B,k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) => ( B = k13_euclid(A) <=> ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => ! [D] : ( m2_finseq_2(D,k1_numbers,k1_euclid(A)) => k1_metric_1(k1_euclid(A),k1_euclid(A),B,C,D) = k12_euclid(k8_euclid(A,C,D)) ) ) ) ) ) ), file(euclid,d6_euclid), []). fof(d7_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ), file(euclid,d7_euclid), []). fof(d8_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ), file(euclid,d8_euclid), []). fof(d8_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) & k1_funct_1(A,B) = k1_funct_1(A,C) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), []). fof(d8_topmetr,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k4_topmetr(A,B) = k5_pcomps_1(k2_topmetr(A,B)) ) ) ), file(topmetr,d8_topmetr), []). fof(d9_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k16_euclid(A) = k5_euclid(A) ) ), file(euclid,d9_euclid), []). fof(d9_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ( m1_pre_topc(B,A) <=> ( r1_tarski(k2_pre_topc(B),k2_pre_topc(A)) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) => ( r2_hidden(C,u1_pre_topc(B)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & r2_hidden(D,u1_pre_topc(A)) & C = k3_xboole_0(D,k2_pre_topc(B)) ) ) ) ) ) ) ) ), file(pre_topc,d9_pre_topc), []). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), []). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), []). fof(dt_k11_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m2_finseq_1(k11_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k11_rvsum_1), []). fof(dt_k12_euclid,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k12_euclid(A),k1_numbers) ) ), file(euclid,k12_euclid), []). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), []). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), []). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), []). fof(dt_k15_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k15_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k15_rvsum_1), []). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_complex1), []). fof(dt_k16_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k16_euclid(A),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k16_euclid), []). fof(dt_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k17_complex1(A),k1_numbers) ) ), file(complex1,k17_complex1), []). fof(dt_k17_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k17_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k17_euclid), []). fof(dt_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k18_complex1(A),k1_numbers) ) ), file(complex1,k18_complex1), []). fof(dt_k18_euclid,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) ) => m1_subset_1(k18_euclid(A,B,C),u1_struct_0(k15_euclid(B))) ) ), file(euclid,k18_euclid), []). fof(dt_k19_euclid,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k19_euclid(A,B),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k19_euclid), []). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_metric_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(C) & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers) & m1_subset_1(D,A) & m1_subset_1(E,B) ) => m1_subset_1(k1_metric_1(A,B,C,D,E),k1_numbers) ) ), file(metric_1,k1_metric_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k1_rcomp_1), []). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_topmetr,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_metric_1(k1_topmetr(A,B)) & m1_topmetr(k1_topmetr(A,B),A) ) ) ), file(topmetr,k1_topmetr), []). fof(dt_k1_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k1_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k1_topreal1), []). fof(dt_k1_tops_1,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k1_tops_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(tops_1,k1_tops_1), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k20_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k20_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k20_euclid), []). fof(dt_k22_borsuk_1,axiom,( l1_pre_topc(k22_borsuk_1) ), file(borsuk_1,k22_borsuk_1), []). fof(dt_k2_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), []). fof(dt_k2_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k2_metric_1), []). fof(dt_k2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k2_pre_topc), []). fof(dt_k2_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k2_rcomp_1), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_topmetr,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( ~ v3_struct_0(k2_topmetr(A,B)) & v1_metric_1(k2_topmetr(A,B)) & m1_topmetr(k2_topmetr(A,B),k8_metric_1) ) ) ), file(topmetr,k2_topmetr), []). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), []). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), []). fof(dt_k3_topmetr,axiom, ( v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & l1_pre_topc(k3_topmetr) ), file(topmetr,k3_topmetr), []). fof(dt_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k3_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k3_topreal1), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k4_euclid(A),k1_numbers) ) ), file(euclid,k4_euclid), []). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), []). fof(dt_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_2(k4_finseqop(A,B,C),A,k4_finseq_2(B,A)) ) ), file(finseqop,k4_finseqop), []). fof(dt_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k4_metric_1), []). fof(dt_k4_pre_topc,axiom,( ! [A,B,C,D] : ( ( l1_struct_0(A) & l1_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => m1_subset_1(k4_pre_topc(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) ), file(pre_topc,k4_pre_topc), []). 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), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k4_topmetr,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( ~ v3_struct_0(k4_topmetr(A,B)) & v1_pre_topc(k4_topmetr(A,B)) & m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ), file(topmetr,k4_topmetr), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ), file(euclid,k5_euclid), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), []). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). fof(dt_k5_topmetr,axiom, ( v1_pre_topc(k5_topmetr) & m1_pre_topc(k5_topmetr,k3_topmetr) ), file(topmetr,k5_topmetr), []). 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), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_rvsum_1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_finseq_1(B,k1_numbers) ) => m2_finseq_1(k7_rvsum_1(A,B),k1_numbers) ) ), file(rvsum_1,k7_rvsum_1), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). fof(dt_k8_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k1_euclid(A)) ) => m2_finseq_2(k8_euclid(A,B,C),k1_numbers,k1_euclid(A)) ) ), file(euclid,k8_euclid), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_metric_1,axiom, ( v1_metric_1(k8_metric_1) & l1_metric_1(k8_metric_1) ), file(metric_1,k8_metric_1), []). fof(dt_k8_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xreal_0(k8_square_1(A)) ) ), file(square_1,k8_square_1), []). fof(dt_k9_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & v1_xreal_0(B) & m1_subset_1(C,k1_euclid(A)) ) => m2_finseq_2(k9_euclid(A,B,C),k1_numbers,k1_euclid(A)) ) ), file(euclid,k9_euclid), []). fof(dt_k9_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & v1_xreal_0(C) ) => m1_subset_1(k9_metric_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(metric_1,k9_metric_1), []). fof(dt_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_1), []). fof(dt_k9_rvsum_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & m1_finseq_1(B,k1_numbers) ) => m2_finseq_1(k9_rvsum_1(A,B),k1_numbers) ) ), file(rvsum_1,k9_rvsum_1), []). fof(dt_k9_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k9_square_1(A),k1_numbers) ) ), file(square_1,k9_square_1), []). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), []). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_m1_connsp_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ! [C] : ( m1_connsp_2(C,A,B) => m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ), file(connsp_2,m1_connsp_2), []). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), []). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_topmetr(B,A) => ( v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) & l1_metric_1(B) ) ) ) ), file(topmetr,m1_topmetr), []). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), []). 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), []). 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), []). fof(dt_u1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), []). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), []). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_connsp_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ? [C] : m1_connsp_2(C,A,B) ) ), file(connsp_2,m1_connsp_2), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), []). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ? [B] : m1_topmetr(B,A) ) ), file(topmetr,m1_topmetr), []). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), 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), []). fof(fc10_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_xreal_0,axiom,( ! [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), []). fof(fc14_xreal_0,axiom,( ! [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), []). fof(fc15_xreal_0,axiom,( ! [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), []). fof(fc16_xreal_0,axiom,( ! [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), []). fof(fc17_xreal_0,axiom,( ! [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), []). fof(fc18_xreal_0,axiom,( ! [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), []). fof(fc19_xreal_0,axiom,( ! [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), []). fof(fc1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k14_euclid(A)) & v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) ) ) ), file(euclid,fc1_euclid), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_topmetr,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k1_topmetr(A,B)) & v1_metric_1(k1_topmetr(A,B)) & v6_metric_1(k1_topmetr(A,B)) & v7_metric_1(k1_topmetr(A,B)) & v8_metric_1(k1_topmetr(A,B)) & v9_metric_1(k1_topmetr(A,B)) ) ) ), file(topmetr,fc1_topmetr), []). fof(fc1_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => ~ v1_xboole_0(k1_topreal1(A,B,C)) ) ), file(topreal1,fc1_topreal1), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc1_xreal_0,axiom,( ! [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), []). fof(fc20_xreal_0,axiom,( ! [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), []). fof(fc21_xreal_0,axiom,( ! [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), []). fof(fc22_xreal_0,axiom,( ! [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), []). fof(fc23_xreal_0,axiom,( ! [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), []). fof(fc24_xreal_0,axiom,( ! [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), []). fof(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc27_xreal_0,axiom,( ! [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), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). fof(fc28_xreal_0,axiom,( ! [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), []). fof(fc29_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), []). fof(fc29_xreal_0,axiom,( ! [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), []). fof(fc2_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) ) ) ), file(euclid,fc2_euclid), []). fof(fc2_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_1), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(nat_1,fc2_nat_1), []). fof(fc2_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(k2_pre_topc(A)) ) ), file(pre_topc,fc2_pre_topc), []). fof(fc2_topmetr,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) ), file(topmetr,fc2_topmetr), []). fof(fc30_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), []). fof(fc30_xreal_0,axiom,( ! [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), []). fof(fc31_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), []). fof(fc32_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), []). fof(fc33_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), []). fof(fc34_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), []). fof(fc35_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), []). fof(fc36_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), []). fof(fc3_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), []). fof(fc3_metric_1,axiom, ( ~ v3_struct_0(k8_metric_1) & v1_metric_1(k8_metric_1) ), file(metric_1,fc3_metric_1), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc3_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc3_pcomps_1), []). fof(fc3_pre_topc,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k3_pre_topc(A,B)) & v1_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc3_pre_topc), []). fof(fc3_topmetr,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & m1_relset_1(B,A,u1_struct_0(k3_topmetr)) ) => ( v1_xcmplx_0(k1_funct_1(B,C)) & v1_xreal_0(k1_funct_1(B,C)) ) ) ), file(topmetr,fc3_topmetr), []). fof(fc3_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), []). fof(fc4_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), []). fof(fc4_metric_1,axiom, ( ~ v3_struct_0(k8_metric_1) & v1_metric_1(k8_metric_1) & v6_metric_1(k8_metric_1) & v7_metric_1(k8_metric_1) & v8_metric_1(k8_metric_1) & v9_metric_1(k8_metric_1) ), file(metric_1,fc4_metric_1), []). fof(fc4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc4_pcomps_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( ~ v3_struct_0(k5_pcomps_1(A)) & v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc4_pcomps_1), []). fof(fc4_pre_topc,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc4_pre_topc), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc4_xreal_0,axiom,( ! [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), []). fof(fc5_borsuk_1,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) ), file(borsuk_1,fc5_borsuk_1), []). fof(fc5_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v4_pre_topc(k2_pre_topc(A),A) ) ), file(pre_topc,fc5_pre_topc), []). fof(fc5_xreal_0,axiom,( ! [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), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fc6_membered,axiom, ( 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), []). fof(fc6_xreal_0,axiom,( ! [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), []). fof(fc7_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_1), []). fof(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). fof(fc8_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), []). fof(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(fraenkel_a_0_4_topreal1,axiom,( ! [A] : ( r2_hidden(A,a_0_4_topreal1) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & A = B & r1_xreal_0(0,B) & r1_xreal_0(B,1) ) ) ), file(topreal1,a_0_4_topreal1), []). fof(fraenkel_a_2_0_rcomp_1,axiom,( ! [A,B,C] : ( ( v1_xreal_0(B) & v1_xreal_0(C) ) => ( r2_hidden(A,a_2_0_rcomp_1(B,C)) <=> ? [D] : ( m1_subset_1(D,k1_numbers) & A = D & r1_xreal_0(B,D) & r1_xreal_0(D,C) ) ) ) ), file(rcomp_1,a_2_0_rcomp_1), []). fof(fraenkel_a_3_0_topreal1,axiom,( ! [A,B,C,D] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) & m1_subset_1(D,u1_struct_0(k15_euclid(B))) ) => ( r2_hidden(A,a_3_0_topreal1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,k1_numbers) & A = k17_euclid(B,k18_euclid(k5_real_1(1,E),B,C),k18_euclid(E,B,D)) & r1_xreal_0(0,E) & r1_xreal_0(E,1) ) ) ) ), file(topreal1,a_3_0_topreal1), []). fof(free_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), []). fof(free_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(involutiveness_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), []). fof(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(l1_topreal1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( u1_struct_0(k14_euclid(A)) = k1_euclid(A) & u1_struct_0(k15_euclid(A)) = k1_euclid(A) ) ) ), file(topreal1,l1_topreal1), []). fof(projectivity_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), []). fof(projectivity_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k17_complex1(A)) = k17_complex1(A) ) ), file(complex1,k17_complex1), []). fof(projectivity_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ), file(complex1,k18_complex1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_metric_1,axiom,( ? [A] : ( l1_metric_1(A) & v1_metric_1(A) ) ), file(metric_1,rc1_metric_1), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ? [B] : ( m1_topmetr(B,A) & v1_metric_1(B) & v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) ) ) ), file(topmetr,rc1_topmetr), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_metric_1,axiom,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) ) ), file(metric_1,rc2_metric_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_pcomps_1,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v3_compts_1(A) ) ), file(pcomps_1,rc2_pcomps_1), []). fof(rc2_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_topmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ? [B] : ( m1_topmetr(B,A) & ~ v3_struct_0(B) & v1_metric_1(B) & v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) ) ) ), file(topmetr,rc2_topmetr), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc2_xreal_0,axiom,( ? [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), []). fof(rc3_metric_1,axiom,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) ) ), file(metric_1,rc3_metric_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_xreal_0,axiom,( ? [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), []). fof(rc4_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), []). fof(rc4_xreal_0,axiom,( ? [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), []). fof(rc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), []). fof(rc5_struct_0,axiom,( ! [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), []). fof(rc6_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc6_pre_topc), []). fof(rc7_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc7_pre_topc), []). fof(redefinition_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(A) = k16_complex1(A) ) ), file(complex1,k17_complex1), []). fof(redefinition_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(A) = k16_complex1(A) ) ), file(complex1,k18_complex1), []). fof(redefinition_k1_metric_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(C) & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers) & m1_subset_1(D,A) & m1_subset_1(E,B) ) => k1_metric_1(A,B,C,D,E) = k1_binop_1(C,D,E) ) ), file(metric_1,k1_metric_1), []). fof(redefinition_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), []). fof(redefinition_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), []). fof(redefinition_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k1_topreal1(A,B,C) ) ), file(topreal1,k3_topreal1), []). fof(redefinition_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_finseqop(A,B,C) = k2_finseq_2(B,C) ) ), file(finseqop,k4_finseqop), []). fof(redefinition_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k2_metric_1(A,B,C) ) ), file(metric_1,k4_metric_1), []). fof(redefinition_k4_pre_topc,axiom,( ! [A,B,C,D] : ( ( l1_struct_0(A) & l1_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => k4_pre_topc(A,B,C,D) = k9_relat_1(C,D) ) ), file(pre_topc,k4_pre_topc), []). fof(redefinition_k4_real_1,axiom,( ! [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), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => k5_euclid(A) = k4_euclid(A) ) ), file(euclid,k5_euclid), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k5_topmetr,axiom,( k5_topmetr = k22_borsuk_1 ), file(topmetr,k5_topmetr), []). fof(redefinition_k6_real_1,axiom,( ! [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), []). fof(redefinition_k8_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k1_euclid(A)) ) => k8_euclid(A,B,C) = k7_rvsum_1(B,C) ) ), file(euclid,k8_euclid), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_k9_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & v1_xreal_0(B) & m1_subset_1(C,k1_euclid(A)) ) => k9_euclid(A,B,C) = k9_rvsum_1(B,C) ) ), file(euclid,k9_euclid), []). fof(redefinition_k9_square_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k9_square_1(A) = k8_square_1(A) ) ), file(square_1,k9_square_1), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_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,B) ) ) ), file(subset_1,m2_subset_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(s1_zfrefle1__e5_19__topreal1,axiom,( ! [A,B,C] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => ( ! [D] : ~ ( r2_hidden(D,k1_rcomp_1(0,1)) & ! [E] : ~ ! [F] : ( m1_subset_1(F,k1_numbers) => ( F = D => E = k17_euclid(A,k18_euclid(k5_real_1(1,F),A,B),k18_euclid(F,A,C)) ) ) ) => ? [D] : ( v1_relat_1(D) & v1_funct_1(D) & k1_relat_1(D) = k1_rcomp_1(0,1) & ! [E] : ( r2_hidden(E,k1_rcomp_1(0,1)) => ! [G] : ( m1_subset_1(G,k1_numbers) => ( G = E => k1_funct_1(D,E) = k17_euclid(A,k18_euclid(k5_real_1(1,G),A,B),k18_euclid(G,A,C)) ) ) ) ) ) ) ), file(topreal1,s1_zfrefle1__e5_19__topreal1), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(t11_heine,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => v2_compts_1(k4_topmetr(A,B)) ) ) ) ), file(heine,t11_heine), []). fof(t11_relset_1,axiom,( ! [A,B,C] : ( v1_relat_1(C) => ( ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) => m2_relset_1(C,A,B) ) ) ), file(relset_1,t11_relset_1), []). fof(t127_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) ) | ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(0,B) ) ) & r1_xreal_0(k7_xcmplx_0(A,B),0) ) ) ) ), file(real_2,t127_real_2), []). fof(t12_metric_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( l1_metric_1(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => ( r2_hidden(D,k9_metric_1(B,C,A)) <=> ( ~ v3_struct_0(B) & ~ r1_xreal_0(A,k2_metric_1(B,C,D)) ) ) ) ) ) ) ), file(metric_1,t12_metric_1), []). fof(t12_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,t12_pre_topc), []). fof(t12_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) ) ), file(relset_1,t12_relset_1), []). fof(t12_topmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m1_topmetr(B,A) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => m1_subset_1(C,u1_struct_0(A)) ) ) ) ), file(topmetr,t12_topmetr), []). fof(t138_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k4_xcmplx_0(A)) = k17_complex1(A) ) ), file(complex1,t138_complex1), []). fof(t14_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => k12_euclid(k9_euclid(A,B,C)) = k4_real_1(k18_complex1(B),k12_euclid(C)) ) ) ) ), file(euclid,t14_euclid), []). fof(t14_topmetr,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => u1_struct_0(k2_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ), file(topmetr,t14_topmetr), []). fof(t15_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ ( ~ r1_xreal_0(B,A) & k2_rcomp_1(A,B) = k1_xboole_0 ) & ( r1_xreal_0(A,B) => ( r2_hidden(A,k1_rcomp_1(A,B)) & r2_hidden(B,k1_rcomp_1(A,B)) ) ) & r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(A,B)) ) ) ) ), file(rcomp_1,t15_rcomp_1), []). fof(t15_topmetr,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k8_metric_1)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k8_metric_1)) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( C = A & D = B ) => k4_metric_1(k8_metric_1,A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) ) ), file(topmetr,t15_topmetr), []). fof(t16_tbsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( v1_xreal_0(C) => ( v6_metric_1(A) => ( r1_xreal_0(C,0) | r2_hidden(B,k9_metric_1(A,B,C)) ) ) ) ) ) ), file(tbsp_1,t16_tbsp_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [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), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), []). fof(t20_topmetr,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) => ( B = C => k3_pre_topc(k15_euclid(A),B) = k5_pcomps_1(k1_topmetr(k14_euclid(A),C)) ) ) ) ) ), file(topmetr,t20_topmetr), []). fof(t21_topmetr,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( ( v9_metric_1(B) & l1_metric_1(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B)))) => ( D = k9_metric_1(B,C,A) => v3_pre_topc(D,k5_pcomps_1(B)) ) ) ) ) ) ), file(topmetr,t21_topmetr), []). fof(t22_topmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ( v3_pre_topc(B,k5_pcomps_1(A)) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ~ ( r2_hidden(C,B) & ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(D,0) & r1_tarski(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ) ), file(topmetr,t22_topmetr), []). fof(t26_compts_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( ( v2_compts_1(A) & v3_compts_1(B) & k1_relat_1(C) = k2_pre_topc(A) & k2_relat_1(C) = k2_pre_topc(B) & v2_funct_1(C) & v5_pre_topc(C,A,B) ) => v3_tops_2(C,A,B) ) ) ) ) ), file(compts_1,t26_compts_1), []). fof(t27_topmetr,axiom,( k4_topmetr(0,1) = k22_borsuk_1 ), file(topmetr,t27_topmetr), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( k2_metric_1(A,B,C) = 0 => B = C ) ) ) <=> v7_metric_1(A) ) ) ), file(metric_1,t2_metric_1), []). fof(t2_real,axiom,( ! [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), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t2_tarski,axiom,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), []). fof(t30_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => k17_euclid(A,k17_euclid(A,B,C),D) = k17_euclid(A,B,k17_euclid(A,C,D)) ) ) ) ) ), file(euclid,t30_euclid), []). fof(t31_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k17_euclid(A,k16_euclid(A),B) = B & k17_euclid(A,B,k16_euclid(A)) = B ) ) ) ), file(euclid,t31_euclid), []). fof(t33_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k18_euclid(1,A,B) = B & k18_euclid(0,A,B) = k16_euclid(A) ) ) ) ), file(euclid,t33_euclid), []). fof(t37_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k18_euclid(k2_xcmplx_0(C,D),A,B) = k17_euclid(A,k18_euclid(C,A,B),k18_euclid(D,A,B)) ) ) ) ) ), file(euclid,t37_euclid), []). fof(t38_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( v1_xreal_0(D) => ~ ( k18_euclid(D,A,B) = k18_euclid(D,A,C) & D != 0 & B != C ) ) ) ) ) ), file(euclid,t38_euclid), []). fof(t38_pcomps_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => v3_compts_1(k5_pcomps_1(A)) ) ), file(pcomps_1,t38_pcomps_1), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). fof(t3_real,axiom,( ! [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), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_topmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m1_pre_topc(B,A) ) => ( v3_compts_1(A) => v3_compts_1(B) ) ) ) ), file(topmetr,t3_topmetr), []). fof(t44_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ( k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(k4_xcmplx_0(C),A,B) & k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(C,A,k19_euclid(A,B)) ) ) ) ) ), file(euclid,t44_euclid), []). fof(t45_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k20_euclid(A,B,C) = k17_euclid(A,B,k19_euclid(A,C)) ) ) ) ), file(euclid,t45_euclid), []). fof(t49_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => k17_euclid(A,B,k20_euclid(A,C,D)) = k20_euclid(A,k17_euclid(A,B,C),D) ) ) ) ) ), file(euclid,t49_euclid), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_real,axiom,( ! [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), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t50_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => k20_euclid(A,B,k17_euclid(A,C,D)) = k20_euclid(A,k20_euclid(A,B,C),D) ) ) ) ) ), file(euclid,t50_euclid), []). fof(t53_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( v1_xreal_0(D) => k18_euclid(D,A,k20_euclid(A,B,C)) = k20_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ), file(euclid,t53_euclid), []). fof(t54_euclid,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k18_euclid(k6_xcmplx_0(C,D),A,B) = k20_euclid(A,k18_euclid(C,A,B),k18_euclid(D,A,B)) ) ) ) ) ), file(euclid,t54_euclid), []). fof(t54_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B,C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(B,k1_tops_1(A,C)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(D,A) & r1_tarski(D,C) & r2_hidden(B,D) ) ) ) ) ), file(tops_1,t54_tops_1), []). fof(t54_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( k7_xcmplx_0(A,B) != 0 => B = k7_xcmplx_0(A,k7_xcmplx_0(A,B)) ) ) ) ), file(xcmplx_1,t54_xcmplx_1), []). fof(t55_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( ( v3_pre_topc(D,B) => k1_tops_1(B,D) = D ) & ( k1_tops_1(A,C) = C => v3_pre_topc(C,A) ) ) ) ) ) ) ), file(tops_1,t55_tops_1), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). fof(t5_metric_1,axiom,( ! [A] : ( ( v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => r1_xreal_0(0,k4_metric_1(A,B,C)) ) ) ) ), file(metric_1,t5_metric_1), []). fof(t5_real,axiom,( ! [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), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [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), []). fof(t70_xreal_1,axiom,( ! [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), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [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), []). fof(t83_borsuk_1,axiom,( u1_struct_0(k22_borsuk_1) = k1_rcomp_1(0,1) ), file(borsuk_1,t83_borsuk_1), []). fof(t88_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ), file(xcmplx_1,t88_xcmplx_1), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [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), []).