% Mizar problem: t104_jordan,jordan,5613,67 fof(t104_jordan,conjecture,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( r1_jordan24(2,A,k23_euclid(k1_real_1(1),0),k23_euclid(1,0)) & r2_subset_1(k3_topreal1(2,k23_euclid(0,3),k23_euclid(0,k1_real_1(3))),A) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_metric_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc10_topreal6,cc10_yellow13,cc11_isomichi,cc11_membered,cc11_topreal6,cc11_yellow13,cc12_isomichi,cc12_membered,cc13_isomichi,cc13_membered,cc14_membered,cc15_membered,cc16_isomichi,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_borsuk_2,cc1_borsuk_4,cc1_funct_1,cc1_funct_2,cc1_jordan,cc1_jordan21,cc1_jordan2c,cc1_membered,cc1_nat_1,cc1_pre_topc,cc1_realset1,cc1_relset_1,cc1_seq_1,cc1_sprect_4,cc1_tex_4,cc1_topreal6,cc1_topreala,cc1_tops_1,cc1_xreal_0,cc1_yellow13,cc20_membered,cc2_borsuk_2,cc2_funct_1,cc2_jordan,cc2_jordan21,cc2_membered,cc2_nat_1,cc2_pscomp_1,cc2_tex_4,cc2_topreal6,cc2_topreala,cc2_tops_1,cc2_xreal_0,cc2_yellow13,cc3_borsuk_2,cc3_jordan,cc3_membered,cc3_nat_1,cc3_pscomp_1,cc3_tex_4,cc3_topreal6,cc3_topreala,cc3_tops_1,cc3_xreal_0,cc3_yellow13,cc4_borsuk_2,cc4_jordan,cc4_membered,cc4_tex_4,cc4_topgrp_1,cc4_topreal6,cc4_topreala,cc4_tops_1,cc4_xreal_0,cc4_yellow13,cc5_funct_2,cc5_jordan,cc5_tex_4,cc5_topreal6,cc5_tops_1,cc5_xreal_0,cc5_yellow13,cc6_funct_2,cc6_membered,cc6_tex_4,cc6_topreal6,cc6_tops_1,cc6_xreal_0,cc6_yellow13,cc7_tex_4,cc7_topreal6,cc7_waybel_3,cc7_xreal_0,cc7_yellow13,cc8_tex_4,cc8_topreal6,cc8_xreal_0,cc8_yellow13,cc9_membered,cc9_topreal6,cc9_yellow13,commutativity_k1_gobrd14,commutativity_k2_struct_0,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_topreal1,commutativity_k3_xboole_0,commutativity_k4_subset_1,commutativity_k5_subset_1,connectedness_r1_xreal_0,d16_euclid,d1_euclid,d1_funct_2,d1_jordan1,d1_jordan24,d20_pscomp_1,d21_pscomp_1,d26_pscomp_1,d2_jgraph_6,d30_pscomp_1,d31_pscomp_1,d32_pscomp_1,d33_pscomp_1,d34_pscomp_1,d35_pscomp_1,d36_pscomp_1,d37_pscomp_1,d38_pscomp_1,d40_pscomp_1,d42_pscomp_1,d43_pscomp_1,d46_pscomp_1,d47_pscomp_1,d5_funct_1,d7_euclid,d8_euclid,d8_jordan6,dt_g1_metric_1,dt_g1_pre_topc,dt_k10_finseq_1,dt_k12_pscomp_1,dt_k13_euclid,dt_k13_pscomp_1,dt_k14_euclid,dt_k15_euclid,dt_k15_pscomp_1,dt_k16_pscomp_1,dt_k17_pscomp_1,dt_k18_pscomp_1,dt_k19_pscomp_1,dt_k1_euclid,dt_k1_funct_1,dt_k1_gobrd14,dt_k1_jordan5c,dt_k1_numbers,dt_k1_pscomp_1,dt_k1_real_1,dt_k1_relat_1,dt_k1_sppol_2,dt_k1_topreal1,dt_k1_topreala,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_pscomp_1,dt_k21_euclid,dt_k21_pscomp_1,dt_k22_borsuk_1,dt_k22_euclid,dt_k22_pscomp_1,dt_k23_euclid,dt_k23_pscomp_1,dt_k24_pscomp_1,dt_k25_pscomp_1,dt_k26_pscomp_1,dt_k28_pscomp_1,dt_k2_funct_2,dt_k2_jgraph_6,dt_k2_jordan5c,dt_k2_relat_1,dt_k2_seq_4,dt_k2_struct_0,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k30_pscomp_1,dt_k31_pscomp_1,dt_k34_pscomp_1,dt_k35_pscomp_1,dt_k3_pre_topc,dt_k3_pscomp_1,dt_k3_real_1,dt_k3_seq_4,dt_k3_topmetr,dt_k3_topreal1,dt_k3_xboole_0,dt_k4_finseq_2,dt_k4_pscomp_1,dt_k4_relset_1,dt_k4_subset_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_subset_1,dt_k5_topmetr,dt_k6_jordan6,dt_k6_real_1,dt_k7_relat_1,dt_k7_xcmplx_0,dt_k8_funct_2,dt_k8_jordan6,dt_k9_jordan6,dt_k9_relat_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_borsuk_2,dt_m1_finseq_2,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,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_borsuk_2,existence_m1_finseq_2,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_pscomp_1,fc10_xreal_0,fc11_pscomp_1,fc11_sppol_2,fc11_xreal_0,fc12_membered,fc12_pscomp_1,fc12_sppol_2,fc12_xreal_0,fc13_membered,fc13_pscomp_1,fc13_xreal_0,fc14_membered,fc14_xreal_0,fc15_jordan,fc15_membered,fc15_pscomp_1,fc16_membered,fc17_topreala,fc1_borsuk_4,fc1_euclid,fc1_jgraph_8,fc1_jordan1a,fc1_jordan21,fc1_jordan22,fc1_jordan5a,fc1_nat_1,fc1_partfun3,fc1_pscomp_1,fc1_seq_1,fc1_struct_0,fc1_subset_1,fc1_topreal6,fc1_waybel_2,fc1_xreal_0,fc1_yellow13,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc27_membered,fc27_xreal_0,fc28_membered,fc28_xreal_0,fc29_membered,fc29_xreal_0,fc2_euclid,fc2_jgraph_8,fc2_jordan,fc2_jordan1a,fc2_jordan21,fc2_jordan22,fc2_jordan2c,fc2_membered,fc2_pscomp_1,fc2_seq_1,fc2_topalg_5,fc2_topmetr,fc2_topreal6,fc2_yellow13,fc30_membered,fc30_xreal_0,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_nat_1,fc3_pre_topc,fc3_pscomp_1,fc3_subset_1,fc3_topalg_5,fc3_topgen_3,fc3_topmetr,fc3_xreal_0,fc4_funct_1,fc4_jordan,fc4_nat_1,fc4_pre_topc,fc4_subset_1,fc4_topalg_5,fc4_topgen_3,fc5_borsuk_1,fc5_jordan,fc5_membered,fc5_topalg_5,fc6_jordan,fc6_membered,fc6_xreal_0,fc7_borsuk_2,fc7_jordan,fc7_topreal6,fc7_xreal_0,fc8_jordan,fc8_xreal_0,fc9_jordan,fc9_pscomp_1,fc9_xreal_0,fraenkel_a_4_3_jgraph_6,free_g1_metric_1,free_g1_pre_topc,idempotence_k2_xboole_0,idempotence_k3_xboole_0,idempotence_k4_subset_1,idempotence_k5_subset_1,involutiveness_k1_real_1,involutiveness_k4_xcmplx_0,irreflexivity_r2_subset_1,l105_jordan,l106_jordan,l110_jordan,l112_jordan,l151_jordan,l152_jordan,l158_jordan,rc10_pscomp_1,rc1_borsuk_2,rc1_borsuk_4,rc1_funct_1,rc1_funct_2,rc1_isomichi,rc1_jordan,rc1_jordan1a,rc1_jordan21,rc1_jordan5a,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_pre_topc,rc1_pscomp_1,rc1_realset1,rc1_seq_1,rc1_subset_1,rc1_tex_4,rc1_topgen_3,rc1_topreal6,rc1_topreala,rc1_tops_1,rc1_xreal_0,rc2_borsuk_1,rc2_borsuk_2,rc2_borsuk_4,rc2_funct_1,rc2_jordan,rc2_jordan1a,rc2_metric_1,rc2_nat_1,rc2_pre_topc,rc2_pscomp_1,rc2_realset1,rc2_subset_1,rc2_tex_4,rc2_topreal6,rc2_tops_1,rc2_xreal_0,rc3_borsuk_1,rc3_borsuk_2,rc3_borsuk_4,rc3_funct_1,rc3_jordan,rc3_jordan1a,rc3_metric_1,rc3_nat_1,rc3_pre_topc,rc3_pscomp_1,rc3_struct_0,rc3_tops_1,rc3_xreal_0,rc4_borsuk_1,rc4_isomichi,rc4_pre_topc,rc4_pscomp_1,rc4_rcomp_3,rc4_tops_1,rc4_waybel_3,rc4_xreal_0,rc4_yellow13,rc5_pre_topc,rc5_rcomp_3,rc5_struct_0,rc5_tops_1,rc6_pre_topc,rc6_topgrp_1,rc7_isomichi,rc7_pre_topc,rc7_pscomp_1,rc8_pscomp_1,rc9_pscomp_1,redefinition_k1_pscomp_1,redefinition_k1_real_1,redefinition_k2_funct_2,redefinition_k2_struct_0,redefinition_k3_pscomp_1,redefinition_k3_real_1,redefinition_k3_topreal1,redefinition_k4_pscomp_1,redefinition_k4_relset_1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k5_subset_1,redefinition_k5_topmetr,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_borsuk_6,redefinition_r2_subset_1,reflexivity_r1_borsuk_6,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,spc3_boole,spc3_numerals,symmetry_r1_borsuk_6,symmetry_r1_xboole_0,symmetry_r2_subset_1,t16_jordan1a,t1_arithm,t1_boole,t1_numerals,t1_real,t1_subset,t1_xboole_1,t2_boole,t2_real,t2_subset,t2_tarski,t39_jordan,t3_real,t3_subset,t3_xboole_0,t4_real,t4_subset,t51_jordan,t52_jordan,t53_topreala,t5_arithm,t5_real,t5_subset,t6_arithm,t6_boole,t6_jgraph_8,t6_real,t7_boole,t7_real,t81_jordan,t89_jordan,t8_boole,t8_real,t90_jordan]), [file(jordan,t104_jordan)]). 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(cc10_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_topreal2(A) => v2_connsp_1(A,k15_euclid(2)) ) ) ), file(topreal6,cc10_topreal6), []). fof(cc10_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v3_compts_1(A) ) => ( v2_pre_topc(A) & v1_urysohn1(A) ) ) ) ), file(yellow13,cc10_yellow13), []). fof(cc11_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_isomichi(B,A) => ( ~ v4_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ) ) ), file(isomichi,cc11_isomichi), []). 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(cc11_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_finset_1(A) => ( v1_membered(A) & v2_membered(A) & v1_rcomp_1(A) ) ) ) ), file(topreal6,cc11_topreal6), []). fof(cc11_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v1_urysohn1(A) ) => ( v2_pre_topc(A) & v2_t_0topsp(A) ) ) ) ), file(yellow13,cc11_yellow13), []). fof(cc12_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v4_isomichi(B,A) => ( ~ v3_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ) ) ), file(isomichi,cc12_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v5_isomichi(B,A) => ( ~ v3_isomichi(B,A) & ~ v4_isomichi(B,A) ) ) ) ) ), file(isomichi,cc13_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_tdlat_3(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( ~ v1_xboole_0(B) & v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_isomichi(B,A) & v4_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ) ) ), file(isomichi,cc16_isomichi), []). 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_borsuk_2,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ( v1_xreal_0(A) & v1_xcmplx_0(A) ) ) ), file(borsuk_2,cc1_borsuk_2), []). fof(cc1_borsuk_4,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_topreal2(A) => ( ~ v1_xboole_0(A) & ~ v1_realset1(A) & v1_topreal2(A) & v2_connsp_1(A,k15_euclid(2)) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) ) ) ) ), file(borsuk_4,cc1_borsuk_4), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_funct_2,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_partfun1(C,A,B) ) => ( v1_funct_1(C) & v1_funct_2(C,A,B) ) ) ) ), file(funct_2,cc1_funct_2), []). fof(cc1_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & v2_sppol_1(A) ) => ( ~ v1_xboole_0(A) & v1_jordan21(A) ) ) ) ), file(jordan,cc1_jordan), []). fof(cc1_jordan21,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_jordan21(A) => ~ v1_xboole_0(A) ) ) ), file(jordan21,cc1_jordan21), []). fof(cc1_jordan2c,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( v6_compts_1(B,k15_euclid(A)) => v1_jordan2c(B,A) ) ) ) ), file(jordan2c,cc1_jordan2c), []). 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_realset1,axiom,( ! [A] : ( ~ v1_realset1(A) => ~ v1_xboole_0(A) ) ), file(realset1,cc1_realset1), []). 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_seq_1,axiom,( ! [A,B] : ( v2_membered(B) => ! [C] : ( m1_relset_1(C,A,B) => ( v1_funct_1(C) => ( v1_funct_1(C) & v1_seq_1(C) ) ) ) ) ), file(seq_1,cc1_seq_1), []). fof(cc1_sprect_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v3_compts_1(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v6_compts_1(B,A) => v4_pre_topc(B,A) ) ) ) ), file(sprect_4,cc1_sprect_4), []). fof(cc1_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( v4_tex_4(B,A) => ( v2_pre_topc(B) & v2_tdlat_3(B) & v3_tdlat_3(B) ) ) ) ) ), file(tex_4,cc1_tex_4), []). fof(cc1_topreal6,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_connsp_1(B,A) ) ) ) ), file(topreal6,cc1_topreal6), []). fof(cc1_topreala,axiom,( ! [A] : ( ( v2_pre_topc(A) & v2_monoid_0(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( v2_pre_topc(B) & v1_monoid_0(B) & v2_monoid_0(B) ) ) ) ), file(topreala,cc1_topreala), []). fof(cc1_tops_1,axiom,( ! [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) => ( v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ) ) ), file(tops_1,cc1_tops_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc1_yellow13,axiom,( ! [A] : ( v1_realset1(A) => v1_finset_1(A) ) ), file(yellow13,cc1_yellow13), []). 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_borsuk_2,axiom,( ! [A,B,C] : ( ( v1_borsuk_2(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ! [D] : ( m1_borsuk_2(D,A,B,C) => ( v1_relat_1(D) & v5_pre_topc(D,k5_topmetr,A) ) ) ) ), file(borsuk_2,cc2_borsuk_2), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_jordan,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( v1_xboole_0(B) => v1_jordan2c(B,A) ) ) ) ), file(jordan,cc2_jordan), []). fof(cc2_jordan21,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_topreal2(A) => ( ~ v1_xboole_0(A) & v4_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & ~ v1_realset1(A) & v1_topreal2(A) & v1_jordan21(A) ) ) ) ), file(jordan21,cc2_jordan21), []). 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_pscomp_1,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v10_pscomp_1(A) ) ) ) ), file(pscomp_1,cc2_pscomp_1), []). fof(cc2_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ~ v2_tdlat_3(B) => ~ v4_tex_4(B,A) ) ) ) ), file(tex_4,cc2_tex_4), []). fof(cc2_topreal6,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v6_compts_1(B,A) ) ) ) ), file(topreal6,cc2_topreal6), []). fof(cc2_topreala,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v3_struct_0(A) & v2_pre_topc(A) ) => ( v2_pre_topc(A) & v1_tdlat_3(A) & v2_tdlat_3(A) ) ) ) ), file(topreala,cc2_topreala), []). fof(cc2_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc2_tops_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc2_yellow13,axiom,( ! [A] : ( l1_struct_0(A) => ( v3_realset2(A) => v6_group_1(A) ) ) ), file(yellow13,cc2_yellow13), []). fof(cc3_borsuk_2,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_connsp_1(A) ) ) ) ), file(borsuk_2,cc3_borsuk_2), []). fof(cc3_jordan,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ~ v1_jordan2c(B,A) => ~ v1_xboole_0(B) ) ) ) ), file(jordan,cc3_jordan), []). 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_pscomp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v10_pscomp_1(A) & l1_pre_topc(A) ) => ! [B] : ( m1_relset_1(B,u1_struct_0(A),k1_numbers) => ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & v9_pscomp_1(B,A) ) => ( ~ v1_xboole_0(B) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & v1_seq_1(B) & v3_seq_2(B) & v1_partfun1(B,u1_struct_0(A),k1_numbers) & v7_pscomp_1(B,u1_struct_0(A)) & v8_pscomp_1(B,u1_struct_0(A)) ) ) ) ) ), file(pscomp_1,cc3_pscomp_1), []). fof(cc3_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ( ~ v3_struct_0(B) & v1_tsep_1(B,A) & v2_tdlat_3(B) ) => ( ~ v3_struct_0(B) & v2_pre_topc(B) & v2_tdlat_3(B) & v3_tdlat_3(B) & v4_tex_4(B,A) ) ) ) ) ), file(tex_4,cc3_tex_4), []). fof(cc3_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ( ~ v1_xboole_0(A) & v1_jordan1(A,2) ) => v2_connsp_1(A,k15_euclid(2)) ) ) ), file(topreal6,cc3_topreal6), []). fof(cc3_topreala,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & v1_tdlat_3(A) & l1_pre_topc(A) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) => ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(C,A,B) ) ) ) ) ), file(topreala,cc3_topreala), []). fof(cc3_tops_1,axiom,( ! [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) => v3_tops_1(B,A) ) ) ) ), file(tops_1,cc3_tops_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(cc3_yellow13,axiom,( ! [A] : ( l1_struct_0(A) => ( ~ v3_realset2(A) => ~ v3_struct_0(A) ) ) ), file(yellow13,cc3_yellow13), []). fof(cc4_borsuk_2,axiom,( ! [A] : ( l1_pre_topc(A) => ( v3_struct_0(A) => v2_t_0topsp(A) ) ) ), file(borsuk_2,cc4_borsuk_2), []). fof(cc4_jordan,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( v1_jordan2c(B,A) => v1_tex_2(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ) ) ), file(jordan,cc4_jordan), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ( ~ v3_struct_0(B) & v1_tsep_1(B,A) & ~ v4_tex_4(B,A) ) => ( ~ v3_struct_0(B) & ~ v3_realset2(B) & v2_pre_topc(B) & ~ v2_tdlat_3(B) & ~ v4_tex_4(B,A) ) ) ) ) ), file(tex_4,cc4_tex_4), []). fof(cc4_topgrp_1,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v3_realset2(A) ) => ( ~ v3_struct_0(A) & v1_topgrp_1(A) ) ) ) ), file(topgrp_1,cc4_topgrp_1), []). fof(cc4_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_sppol_1(A) => ~ v1_xboole_0(A) ) ) ), file(topreal6,cc4_topreal6), []). fof(cc4_topreala,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & l1_pre_topc(B) ) => ! [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) => ( ( v1_funct_1(C) & v1_xboole_0(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(C,A,B) ) ) ) ) ), file(topreala,cc4_topreala), []). fof(cc4_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_tops_1(B,A) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc4_tops_1), []). 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(cc4_yellow13,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v4_pre_topc(B,A) ) ) ) ), file(yellow13,cc4_yellow13), []). fof(cc5_funct_2,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc5_funct_2), []). fof(cc5_jordan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v3_compts_1(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ~ v3_struct_0(B) => ( ~ v3_struct_0(B) & v2_pre_topc(B) & v3_compts_1(B) ) ) ) ) ), file(jordan,cc5_jordan), []). fof(cc5_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ( ~ v3_struct_0(B) & v2_tdlat_3(B) & ~ v4_tex_4(B,A) ) => ( ~ v3_struct_0(B) & v2_pre_topc(B) & ~ v1_tsep_1(B,A) & v2_tex_2(B,A) ) ) ) ) ), file(tex_4,cc5_tex_4), []). fof(cc5_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v2_sppol_1(A) => ~ v1_xboole_0(A) ) ) ), file(topreal6,cc5_topreal6), []). fof(cc5_tops_1,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) & v2_tops_1(B,A) ) => ( v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ) ) ), file(tops_1,cc5_tops_1), []). 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(cc5_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & v6_group_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_tdlat_3(A) & v3_tdlat_3(A) & v4_tdlat_3(A) & v5_tdlat_3(A) ) ) ) ), file(yellow13,cc5_yellow13), []). fof(cc6_funct_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ! [C] : ( m1_relset_1(C,A,B) => ( ( v1_funct_1(C) & v1_funct_2(C,A,B) ) => ( v1_funct_1(C) & ~ v1_xboole_0(C) & v1_partfun1(C,A,B) & v1_funct_2(C,A,B) ) ) ) ) ), file(funct_2,cc6_funct_2), []). 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_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ( ~ v3_struct_0(B) & v1_borsuk_1(B,A) & v2_tdlat_3(B) ) => ( ~ v3_struct_0(B) & v2_pre_topc(B) & v2_tdlat_3(B) & v3_tdlat_3(B) & v4_tex_4(B,A) ) ) ) ) ), file(tex_4,cc6_tex_4), []). fof(cc6_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v2_topreal4(A,k15_euclid(2)) => ( v3_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) ) ) ) ), file(topreal6,cc6_topreal6), []). fof(cc6_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v3_pre_topc(B,A) & v3_tops_1(B,A) ) => ( v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ) ) ), file(tops_1,cc6_tops_1), []). 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(cc6_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v6_group_1(A) ) => ( v2_pre_topc(A) & v2_compts_1(A) ) ) ) ), file(yellow13,cc6_yellow13), []). fof(cc7_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ( ~ v3_struct_0(B) & v1_borsuk_1(B,A) & ~ v4_tex_4(B,A) ) => ( ~ v3_struct_0(B) & ~ v3_realset2(B) & v2_pre_topc(B) & ~ v2_tdlat_3(B) & ~ v4_tex_4(B,A) ) ) ) ) ), file(tex_4,cc7_tex_4), []). fof(cc7_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ( v3_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) ) => v2_topreal4(A,k15_euclid(2)) ) ) ), file(topreal6,cc7_topreal6), []). fof(cc7_waybel_3,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) & v3_compts_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v4_compts_1(A) & v5_compts_1(A) & v6_waybel_3(A) ) ) ) ), file(waybel_3,cc7_waybel_3), []). 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(cc7_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_tdlat_3(A) ) => ( v2_pre_topc(A) & v1_urysohn1(A) & v3_compts_1(A) & v4_compts_1(A) & v5_compts_1(A) ) ) ) ), file(yellow13,cc7_yellow13), []). fof(cc8_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( ( ~ v3_struct_0(B) & v2_tdlat_3(B) & ~ v4_tex_4(B,A) ) => ( ~ v3_struct_0(B) & v2_pre_topc(B) & ~ v1_borsuk_1(B,A) & v2_tex_2(B,A) ) ) ) ) ), file(tex_4,cc8_tex_4), []). fof(cc8_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_xboole_0(A) => v1_sppol_1(A) ) ) ), file(topreal6,cc8_topreal6), []). 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(cc8_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & v5_compts_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v4_compts_1(A) ) ) ) ), file(yellow13,cc8_yellow13), []). 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(cc9_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_xboole_0(A) => v2_sppol_1(A) ) ) ), file(topreal6,cc9_topreal6), []). fof(cc9_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v1_urysohn1(A) & v4_compts_1(A) ) => ( v2_pre_topc(A) & v3_compts_1(A) ) ) ) ), file(yellow13,cc9_yellow13), []). fof(commutativity_k1_gobrd14,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))) ) => k1_gobrd14(A,B,C) = k1_gobrd14(A,C,B) ) ), file(gobrd14,k1_gobrd14), []). fof(commutativity_k2_struct_0,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k2_struct_0(A,B,C) = k2_struct_0(A,C,B) ) ), file(struct_0,k2_struct_0), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). 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_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), []). fof(commutativity_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_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(d16_euclid,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k23_euclid(A,B) = k10_finseq_1(A,B) ) ) ), file(euclid,d16_euclid), []). 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_jordan1,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( v1_jordan1(B,A) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( ( r2_hidden(C,B) & r2_hidden(D,B) ) => r1_tarski(k3_topreal1(A,C,D),B) ) ) ) ) ) ) ), file(jordan1,d1_jordan1), []). fof(d1_jordan24,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] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( r1_jordan24(A,B,C,D) <=> ( r2_hidden(C,B) & r2_hidden(D,B) & ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ( ( r2_hidden(E,B) & r2_hidden(F,B) ) => r1_xreal_0(k1_gobrd14(A,E,F),k1_gobrd14(A,C,D)) ) ) ) ) ) ) ) ) ) ), file(jordan24,d1_jordan24), []). fof(d20_pscomp_1,axiom,( ! [A] : ( l1_struct_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m2_relset_1(B,u1_struct_0(A),k1_numbers) ) => k12_pscomp_1(A,B) = k4_pscomp_1(k2_funct_2(u1_struct_0(A),k1_numbers,B,u1_struct_0(A))) ) ) ), file(pscomp_1,d20_pscomp_1), []). fof(d21_pscomp_1,axiom,( ! [A] : ( l1_struct_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m2_relset_1(B,u1_struct_0(A),k1_numbers) ) => k13_pscomp_1(A,B) = k3_pscomp_1(k2_funct_2(u1_struct_0(A),k1_numbers,B,u1_struct_0(A))) ) ) ), file(pscomp_1,d21_pscomp_1), []). fof(d26_pscomp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m2_relset_1(C,u1_struct_0(A),k1_numbers) ) => k15_pscomp_1(A,B,C) = k7_relat_1(C,B) ) ) ) ), file(pscomp_1,d26_pscomp_1), []). fof(d2_jgraph_6,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k2_jgraph_6(A,B,C,D) = a_4_3_jgraph_6(A,B,C,D) ) ) ) ) ), file(jgraph_6,d2_jgraph_6), []). fof(d30_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k18_pscomp_1(A) = k12_pscomp_1(k3_pre_topc(k15_euclid(2),A),k15_pscomp_1(k15_euclid(2),A,k16_pscomp_1)) ) ), file(pscomp_1,d30_pscomp_1), []). fof(d31_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k19_pscomp_1(A) = k13_pscomp_1(k3_pre_topc(k15_euclid(2),A),k15_pscomp_1(k15_euclid(2),A,k17_pscomp_1)) ) ), file(pscomp_1,d31_pscomp_1), []). fof(d32_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k20_pscomp_1(A) = k13_pscomp_1(k3_pre_topc(k15_euclid(2),A),k15_pscomp_1(k15_euclid(2),A,k16_pscomp_1)) ) ), file(pscomp_1,d32_pscomp_1), []). fof(d33_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k21_pscomp_1(A) = k12_pscomp_1(k3_pre_topc(k15_euclid(2),A),k15_pscomp_1(k15_euclid(2),A,k17_pscomp_1)) ) ), file(pscomp_1,d33_pscomp_1), []). fof(d34_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k22_pscomp_1(A) = k23_euclid(k18_pscomp_1(A),k21_pscomp_1(A)) ) ), file(pscomp_1,d34_pscomp_1), []). fof(d35_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k23_pscomp_1(A) = k23_euclid(k18_pscomp_1(A),k19_pscomp_1(A)) ) ), file(pscomp_1,d35_pscomp_1), []). fof(d36_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k24_pscomp_1(A) = k23_euclid(k20_pscomp_1(A),k19_pscomp_1(A)) ) ), file(pscomp_1,d36_pscomp_1), []). fof(d37_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k25_pscomp_1(A) = k23_euclid(k20_pscomp_1(A),k21_pscomp_1(A)) ) ), file(pscomp_1,d37_pscomp_1), []). fof(d38_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k26_pscomp_1(A) = k5_subset_1(u1_struct_0(k15_euclid(2)),k3_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),A) ) ), file(pscomp_1,d38_pscomp_1), []). fof(d40_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k28_pscomp_1(A) = k5_subset_1(u1_struct_0(k15_euclid(2)),k3_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),A) ) ), file(pscomp_1,d40_pscomp_1), []). fof(d42_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k30_pscomp_1(A) = k23_euclid(k18_pscomp_1(A),k12_pscomp_1(k3_pre_topc(k15_euclid(2),k26_pscomp_1(A)),k15_pscomp_1(k15_euclid(2),k26_pscomp_1(A),k17_pscomp_1))) ) ), file(pscomp_1,d42_pscomp_1), []). fof(d43_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k31_pscomp_1(A) = k23_euclid(k18_pscomp_1(A),k13_pscomp_1(k3_pre_topc(k15_euclid(2),k26_pscomp_1(A)),k15_pscomp_1(k15_euclid(2),k26_pscomp_1(A),k17_pscomp_1))) ) ), file(pscomp_1,d43_pscomp_1), []). fof(d46_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k34_pscomp_1(A) = k23_euclid(k20_pscomp_1(A),k13_pscomp_1(k3_pre_topc(k15_euclid(2),k28_pscomp_1(A)),k15_pscomp_1(k15_euclid(2),k28_pscomp_1(A),k17_pscomp_1))) ) ), file(pscomp_1,d46_pscomp_1), []). fof(d47_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k35_pscomp_1(A) = k23_euclid(k20_pscomp_1(A),k12_pscomp_1(k3_pre_topc(k15_euclid(2),k28_pscomp_1(A)),k15_pscomp_1(k15_euclid(2),k28_pscomp_1(A),k17_pscomp_1))) ) ), file(pscomp_1,d47_pscomp_1), []). 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(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_jordan6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_topreal2(A) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( B = k8_jordan6(A) <=> ( r1_topreal1(k15_euclid(2),k30_pscomp_1(A),k34_pscomp_1(A),B) & ? [C] : ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & r1_topreal1(k15_euclid(2),k34_pscomp_1(A),k30_pscomp_1(A),C) & k5_subset_1(u1_struct_0(k15_euclid(2)),B,C) = k2_struct_0(k15_euclid(2),k30_pscomp_1(A),k34_pscomp_1(A)) & k4_subset_1(u1_struct_0(k15_euclid(2)),B,C) = A & ~ r1_xreal_0(k22_euclid(k1_jordan5c(B,k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),2)),k30_pscomp_1(A),k34_pscomp_1(A))),k22_euclid(k2_jordan5c(C,k6_jordan6(k6_real_1(k3_real_1(k18_pscomp_1(A),k20_pscomp_1(A)),2)),k34_pscomp_1(A),k30_pscomp_1(A)))) ) ) ) ) ) ) ), file(jordan6,d8_jordan6), []). 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_k10_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), []). fof(dt_k12_pscomp_1,axiom,( ! [A,B] : ( ( l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m1_relset_1(B,u1_struct_0(A),k1_numbers) ) => m1_subset_1(k12_pscomp_1(A,B),k1_numbers) ) ), file(pscomp_1,k12_pscomp_1), []). 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_k13_pscomp_1,axiom,( ! [A,B] : ( ( l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m1_relset_1(B,u1_struct_0(A),k1_numbers) ) => m1_subset_1(k13_pscomp_1(A,B),k1_numbers) ) ), file(pscomp_1,k13_pscomp_1), []). 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_pscomp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m1_relset_1(C,u1_struct_0(A),k1_numbers) ) => ( v1_funct_1(k15_pscomp_1(A,B,C)) & v1_funct_2(k15_pscomp_1(A,B,C),u1_struct_0(k3_pre_topc(A,B)),k1_numbers) & m2_relset_1(k15_pscomp_1(A,B,C),u1_struct_0(k3_pre_topc(A,B)),k1_numbers) ) ) ), file(pscomp_1,k15_pscomp_1), []). fof(dt_k16_pscomp_1,axiom, ( v1_funct_1(k16_pscomp_1) & v1_funct_2(k16_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) & m2_relset_1(k16_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) ), file(pscomp_1,k16_pscomp_1), []). fof(dt_k17_pscomp_1,axiom, ( v1_funct_1(k17_pscomp_1) & v1_funct_2(k17_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) & m2_relset_1(k17_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) ), file(pscomp_1,k17_pscomp_1), []). fof(dt_k18_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k18_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k18_pscomp_1), []). fof(dt_k19_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k19_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k19_pscomp_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_gobrd14,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_gobrd14(A,B,C),k1_numbers) ) ), file(gobrd14,k1_gobrd14), []). fof(dt_k1_jordan5c,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(C,u1_struct_0(k15_euclid(2))) & m1_subset_1(D,u1_struct_0(k15_euclid(2))) ) => m1_subset_1(k1_jordan5c(A,B,C,D),u1_struct_0(k15_euclid(2))) ) ), file(jordan5c,k1_jordan5c), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_pscomp_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => m1_subset_1(k1_pscomp_1(A,B,C),k1_zfmisc_1(B)) ) ), file(pscomp_1,k1_pscomp_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_sppol_2,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => m1_subset_1(k1_sppol_2(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(sppol_2,k1_sppol_2), []). 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_topreala,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => m1_pre_topc(k1_topreala(A,B,C,D),k15_euclid(2)) ) ), file(topreala,k1_topreala), []). 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_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k20_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k20_pscomp_1), []). fof(dt_k21_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k21_euclid(A),k1_numbers) ) ), file(euclid,k21_euclid), []). fof(dt_k21_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k21_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k21_pscomp_1), []). fof(dt_k22_borsuk_1,axiom,( l1_pre_topc(k22_borsuk_1) ), file(borsuk_1,k22_borsuk_1), []). fof(dt_k22_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k22_euclid(A),k1_numbers) ) ), file(euclid,k22_euclid), []). fof(dt_k22_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k22_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k22_pscomp_1), []). fof(dt_k23_euclid,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k23_euclid(A,B),u1_struct_0(k15_euclid(2))) ) ), file(euclid,k23_euclid), []). fof(dt_k23_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k23_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k23_pscomp_1), []). fof(dt_k24_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k24_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k24_pscomp_1), []). fof(dt_k25_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k25_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k25_pscomp_1), []). fof(dt_k26_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k26_pscomp_1(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(pscomp_1,k26_pscomp_1), []). fof(dt_k28_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k28_pscomp_1(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(pscomp_1,k28_pscomp_1), []). fof(dt_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ), file(funct_2,k2_funct_2), []). fof(dt_k2_jgraph_6,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => m1_subset_1(k2_jgraph_6(A,B,C,D),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(jgraph_6,k2_jgraph_6), []). fof(dt_k2_jordan5c,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(C,u1_struct_0(k15_euclid(2))) & m1_subset_1(D,u1_struct_0(k15_euclid(2))) ) => m1_subset_1(k2_jordan5c(A,B,C,D),u1_struct_0(k15_euclid(2))) ) ), file(jordan5c,k2_jordan5c), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k2_seq_4(A)) ) ), file(seq_4,k2_seq_4), []). fof(dt_k2_struct_0,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_struct_0(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k2_struct_0), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). 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_k30_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k30_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k30_pscomp_1), []). fof(dt_k31_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k31_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k31_pscomp_1), []). fof(dt_k34_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k34_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k34_pscomp_1), []). fof(dt_k35_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => m1_subset_1(k35_pscomp_1(A),u1_struct_0(k15_euclid(2))) ) ), file(pscomp_1,k35_pscomp_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_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k3_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k3_pscomp_1), []). 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_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k3_seq_4(A)) ) ), file(seq_4,k3_seq_4), []). 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_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_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k4_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k4_pscomp_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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), []). 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_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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_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_jordan6,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k6_jordan6(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ), file(jordan6,k6_jordan6), []). 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_k7_relat_1,axiom,( ! [A,B] : ( v1_relat_1(A) => v1_relat_1(k7_relat_1(A,B)) ) ), file(relat_1,k7_relat_1), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). 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_jordan6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k8_jordan6(A)) & m1_subset_1(k8_jordan6(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ) ), file(jordan6,k8_jordan6), []). fof(dt_k9_jordan6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k9_jordan6(A)) & m1_subset_1(k9_jordan6(A),k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) ) ), file(jordan6,k9_jordan6), []). fof(dt_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_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_borsuk_2,axiom,( ! [A,B,C] : ( ( l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ! [D] : ( m1_borsuk_2(D,A,B,C) => ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) ) ) ), file(borsuk_2,m1_borsuk_2), []). 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_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_borsuk_2,axiom,( ! [A,B,C] : ( ( l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ? [D] : m1_borsuk_2(D,A,B,C) ) ), file(borsuk_2,m1_borsuk_2), []). 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_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_pscomp_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & v6_compts_1(B,A) & 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)) & v2_pre_topc(k3_pre_topc(A,B)) & v2_compts_1(k3_pre_topc(A,B)) ) ) ), file(pscomp_1,fc10_pscomp_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_pscomp_1,axiom, ( ~ v1_xboole_0(k16_pscomp_1) & v1_relat_1(k16_pscomp_1) & v1_funct_1(k16_pscomp_1) & v1_funct_2(k16_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) & v1_seq_1(k16_pscomp_1) & v1_partfun1(k16_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) & v9_pscomp_1(k16_pscomp_1,k15_euclid(2)) ), file(pscomp_1,fc11_pscomp_1), []). fof(fc11_sppol_2,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)) & v6_compts_1(k1_topreal1(A,B,C),k15_euclid(A)) ) ) ), file(sppol_2,fc11_sppol_2), []). 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_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc12_pscomp_1,axiom, ( ~ v1_xboole_0(k17_pscomp_1) & v1_relat_1(k17_pscomp_1) & v1_funct_1(k17_pscomp_1) & v1_funct_2(k17_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) & v1_seq_1(k17_pscomp_1) & v1_partfun1(k17_pscomp_1,u1_struct_0(k15_euclid(2)),k1_numbers) & v9_pscomp_1(k17_pscomp_1,k15_euclid(2)) ), file(pscomp_1,fc12_pscomp_1), []). fof(fc12_sppol_2,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => ( ~ v1_xboole_0(k1_sppol_2(A,B,C,D)) & v6_compts_1(k1_sppol_2(A,B,C,D),k15_euclid(2)) ) ) ), file(sppol_2,fc12_sppol_2), []). 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_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(fc13_pscomp_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k26_pscomp_1(A)) & v6_compts_1(k26_pscomp_1(A),k15_euclid(2)) ) ) ), file(pscomp_1,fc13_pscomp_1), []). 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_membered,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), []). 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_jordan,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => ( v4_pre_topc(k2_jgraph_6(A,B,C,D),k15_euclid(2)) & v1_jordan1(k2_jgraph_6(A,B,C,D),2) ) ) ), file(jordan,fc15_jordan), []). fof(fc15_membered,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), []). fof(fc15_pscomp_1,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k28_pscomp_1(A)) & v6_compts_1(k28_pscomp_1(A),k15_euclid(2)) ) ) ), file(pscomp_1,fc15_pscomp_1), []). fof(fc16_membered,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), []). fof(fc17_topreala,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) & v1_xreal_0(C) & ~ v3_xreal_0(C) & v1_xreal_0(D) & ~ v3_xreal_0(D) ) => ( ~ v3_struct_0(k1_topreala(A,C,B,D)) & v2_pre_topc(k1_topreala(A,C,B,D)) & v1_monoid_0(k1_topreala(A,C,B,D)) & v2_monoid_0(k1_topreala(A,C,B,D)) ) ) ), file(topreala,fc17_topreala), []). fof(fc1_borsuk_4,axiom, ( ~ v1_xboole_0(u1_struct_0(k5_topmetr)) & v1_membered(u1_struct_0(k5_topmetr)) & v2_membered(u1_struct_0(k5_topmetr)) ), file(borsuk_4,fc1_borsuk_4), []). 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_jgraph_8,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => v1_jordan1(k2_jgraph_6(A,B,C,D),2) ) ), file(jgraph_8,fc1_jgraph_8), []). fof(fc1_jordan1a,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k9_jordan6(A)) & v4_pre_topc(k9_jordan6(A),k15_euclid(2)) & v6_compts_1(k9_jordan6(A),k15_euclid(2)) & v1_jordan2c(k9_jordan6(A),2) ) ) ), file(jordan1a,fc1_jordan1a), []). fof(fc1_jordan21,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k9_jordan6(A)) & v4_pre_topc(k9_jordan6(A),k15_euclid(2)) & v1_jordan2c(k9_jordan6(A),2) & v6_compts_1(k9_jordan6(A),k15_euclid(2)) & v1_jordan21(k9_jordan6(A)) ) ) ), file(jordan21,fc1_jordan21), []). fof(fc1_jordan22,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k8_jordan6(A)) & v4_pre_topc(k8_jordan6(A),k15_euclid(2)) & v2_connsp_1(k8_jordan6(A),k15_euclid(2)) & v1_jordan2c(k8_jordan6(A),2) & v6_compts_1(k8_jordan6(A),k15_euclid(2)) & v1_jordan21(k8_jordan6(A)) ) ) ), file(jordan22,fc1_jordan22), []). fof(fc1_jordan5a,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)) & v3_compts_1(k15_euclid(A)) & v1_borsuk_2(k15_euclid(A)) ) ) ), file(jordan5a,fc1_jordan5a), []). 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_partfun3,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k7_xcmplx_0(A,A)) & v1_xreal_0(k7_xcmplx_0(A,A)) & ~ v3_xreal_0(k7_xcmplx_0(A,A)) ) ) ), file(partfun3,fc1_partfun3), []). fof(fc1_pscomp_1,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) & ~ v1_setfam_1(k1_numbers) ), file(pscomp_1,fc1_pscomp_1), []). fof(fc1_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_xcmplx_0(k1_funct_1(A,B)) & v1_xreal_0(k1_funct_1(A,B)) ) ) ), file(seq_1,fc1_seq_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_topreal6,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & v1_finset_1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v6_group_1(k3_pre_topc(A,B)) ) ) ), file(topreal6,fc1_topreal6), []). fof(fc1_waybel_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & ~ v1_xboole_0(D) & m1_subset_1(D,k1_zfmisc_1(A)) ) => ~ v1_xboole_0(k9_relat_1(C,D)) ) ), file(waybel_2,fc1_waybel_2), []). 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(fc1_yellow13,axiom,( ! [A] : ( ( v6_group_1(A) & l1_struct_0(A) ) => v1_finset_1(u1_struct_0(A)) ) ), file(yellow13,fc1_yellow13), []). fof(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc23_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), []). fof(fc24_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), []). fof(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc26_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), []). 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_jgraph_8,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) ) => ( v2_pre_topc(k1_topreala(A,B,C,D)) & v1_topalg_2(k1_topreala(A,B,C,D),2) ) ) ), file(jgraph_8,fc2_jgraph_8), []). fof(fc2_jordan,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_xreal_0(k1_gobrd14(A,B,C)) & ~ v3_xreal_0(k1_gobrd14(A,B,C)) & v1_xcmplx_0(k1_gobrd14(A,B,C)) ) ) ), file(jordan,fc2_jordan), []). fof(fc2_jordan1a,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k8_jordan6(A)) & v4_pre_topc(k8_jordan6(A),k15_euclid(2)) & v6_compts_1(k8_jordan6(A),k15_euclid(2)) & v1_jordan2c(k8_jordan6(A),2) ) ) ), file(jordan1a,fc2_jordan1a), []). fof(fc2_jordan21,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k8_jordan6(A)) & v4_pre_topc(k8_jordan6(A),k15_euclid(2)) & v1_jordan2c(k8_jordan6(A),2) & v6_compts_1(k8_jordan6(A),k15_euclid(2)) & v1_jordan21(k8_jordan6(A)) ) ) ), file(jordan21,fc2_jordan21), []). fof(fc2_jordan22,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k9_jordan6(A)) & v4_pre_topc(k9_jordan6(A),k15_euclid(2)) & v2_connsp_1(k9_jordan6(A),k15_euclid(2)) & v1_jordan2c(k9_jordan6(A),2) & v6_compts_1(k9_jordan6(A),k15_euclid(2)) & v1_jordan21(k9_jordan6(A)) ) ) ), file(jordan22,fc2_jordan22), []). fof(fc2_jordan2c,axiom,( ! [A,B] : ( ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k1_topreal1(2,A,B)) & v2_tops_1(k1_topreal1(2,A,B),k15_euclid(2)) ) ) ), file(jordan2c,fc2_jordan2c), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_pscomp_1,axiom, ( ~ v1_xboole_0(k5_ordinal2) & v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) & ~ v1_setfam_1(k5_ordinal2) ), file(pscomp_1,fc2_pscomp_1), []). fof(fc2_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_seq_1(k7_relat_1(A,B)) ) ) ), file(seq_1,fc2_seq_1), []). fof(fc2_topalg_5,axiom,( ! [A] : ( ( ~ v6_group_1(A) & l1_struct_0(A) ) => ~ v1_finset_1(u1_struct_0(A)) ) ), file(topalg_5,fc2_topalg_5), []). fof(fc2_topmetr,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) ), file(topmetr,fc2_topmetr), []). fof(fc2_topreal6,axiom,( ! [A,B] : ( ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & m1_subset_1(B,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k1_topreal1(2,A,B)) & v2_connsp_1(k1_topreal1(2,A,B),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,A,B),2) & v2_tops_1(k1_topreal1(2,A,B),k15_euclid(2)) ) ) ), file(topreal6,fc2_topreal6), []). fof(fc2_yellow13,axiom,( ! [A] : ( ( v3_realset2(A) & l1_struct_0(A) ) => v1_realset1(u1_struct_0(A)) ) ), file(yellow13,fc2_yellow13), []). 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_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_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_pscomp_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => ~ v1_xboole_0(k9_relat_1(C,A)) ) ), file(pscomp_1,fc3_pscomp_1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_topalg_5,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v3_pre_topc(B,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)) & v1_tsep_1(k3_pre_topc(A,B),A) ) ) ), file(topalg_5,fc3_topalg_5), []). fof(fc3_topgen_3,axiom, ( ~ v1_xboole_0(k1_numbers) & ~ v1_finset_1(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(topgen_3,fc3_topgen_3), []). 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_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) ) ) ), file(funct_1,fc4_funct_1), []). fof(fc4_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A))) & v4_pre_topc(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v6_compts_1(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v1_jordan2c(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),2) & v2_connsp_1(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v2_tops_1(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v3_tops_1(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A)),2) & v1_sppol_1(k1_topreal1(2,k22_pscomp_1(A),k25_pscomp_1(A))) ) ) ), file(jordan,fc4_jordan), []). 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_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_topalg_5,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v1_connsp_2(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) ), file(topalg_5,fc4_topalg_5), []). fof(fc4_topgen_3,axiom,( ! [A] : ( ~ v1_finset_1(A) => ( ~ v1_xboole_0(k1_zfmisc_1(A)) & ~ v1_finset_1(k1_zfmisc_1(A)) ) ) ), file(topgen_3,fc4_topgen_3), []). 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_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A))) & v4_pre_topc(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v6_compts_1(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v1_jordan2c(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),2) & v2_connsp_1(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v2_tops_1(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v3_tops_1(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A)),2) & v2_sppol_1(k1_topreal1(2,k23_pscomp_1(A),k22_pscomp_1(A))) ) ) ), file(jordan,fc5_jordan), []). 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_topalg_5,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v3_pre_topc(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ( ~ v3_struct_0(k3_pre_topc(k5_topmetr,A)) & v1_pre_topc(k3_pre_topc(k5_topmetr,A)) & v2_pre_topc(k3_pre_topc(k5_topmetr,A)) & v1_connsp_2(k3_pre_topc(k5_topmetr,A)) & v1_tsep_1(k3_pre_topc(k5_topmetr,A),k5_topmetr) ) ) ), file(topalg_5,fc5_topalg_5), []). fof(fc6_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A))) & v4_pre_topc(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v6_compts_1(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v1_jordan2c(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),2) & v2_connsp_1(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v2_tops_1(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v3_tops_1(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A)),2) & v2_sppol_1(k1_topreal1(2,k24_pscomp_1(A),k25_pscomp_1(A))) ) ) ), file(jordan,fc6_jordan), []). 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_borsuk_2,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) ), file(borsuk_2,fc7_borsuk_2), []). fof(fc7_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A))) & v4_pre_topc(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v6_compts_1(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v1_jordan2c(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),2) & v2_connsp_1(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v2_tops_1(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v3_tops_1(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A)),2) & v1_sppol_1(k1_topreal1(2,k25_pscomp_1(A),k22_pscomp_1(A))) ) ) ), file(jordan,fc7_jordan), []). fof(fc7_topreal6,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & v3_compts_1(k3_topmetr) ), file(topreal6,fc7_topreal6), []). 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_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A))) & v4_pre_topc(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),k15_euclid(2)) & v6_compts_1(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),k15_euclid(2)) & v1_jordan2c(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),2) & v2_connsp_1(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),k15_euclid(2)) & v2_tops_1(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),k15_euclid(2)) & v3_tops_1(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A)),2) & v2_sppol_1(k1_topreal1(2,k22_pscomp_1(A),k23_pscomp_1(A))) ) ) ), file(jordan,fc8_jordan), []). 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_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_xboole_0(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A))) & v4_pre_topc(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),k15_euclid(2)) & v6_compts_1(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),k15_euclid(2)) & v1_jordan2c(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),2) & v2_connsp_1(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),k15_euclid(2)) & v2_tops_1(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),k15_euclid(2)) & v3_tops_1(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),k15_euclid(2)) & v1_jordan1(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A)),2) & v2_sppol_1(k1_topreal1(2,k25_pscomp_1(A),k24_pscomp_1(A))) ) ) ), file(jordan,fc9_jordan), []). fof(fc9_pscomp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & v9_pscomp_1(B,A) & m1_relset_1(B,u1_struct_0(A),k1_numbers) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_relat_1(k15_pscomp_1(A,C,B)) & v1_funct_1(k15_pscomp_1(A,C,B)) & v1_funct_2(k15_pscomp_1(A,C,B),u1_struct_0(k3_pre_topc(A,C)),k1_numbers) & v1_seq_1(k15_pscomp_1(A,C,B)) & v1_partfun1(k15_pscomp_1(A,C,B),u1_struct_0(k3_pre_topc(A,C)),k1_numbers) & v9_pscomp_1(k15_pscomp_1(A,C,B),k3_pre_topc(A,C)) ) ) ), file(pscomp_1,fc9_pscomp_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_4_3_jgraph_6,axiom,( ! [A,B,C,D,E] : ( ( v1_xreal_0(B) & v1_xreal_0(C) & v1_xreal_0(D) & v1_xreal_0(E) ) => ( r2_hidden(A,a_4_3_jgraph_6(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(2))) & A = F & r1_xreal_0(B,k21_euclid(F)) & r1_xreal_0(k21_euclid(F),C) & r1_xreal_0(D,k22_euclid(F)) & r1_xreal_0(k22_euclid(F),E) ) ) ) ), file(jgraph_6,a_4_3_jgraph_6), []). 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_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(idempotence_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), []). fof(idempotence_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), []). 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(irreflexivity_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r2_subset_1(A,A) ) ), file(subset_1,r2_subset_1), []). fof(l105_jordan,axiom,( k21_euclid(k23_euclid(k1_real_1(1),0)) = k1_real_1(1) ), file(jordan,l105_jordan), []). fof(l106_jordan,axiom,( k21_euclid(k23_euclid(1,0)) = 1 ), file(jordan,l106_jordan), []). fof(l110_jordan,axiom,( k22_euclid(k23_euclid(0,3)) = 3 ), file(jordan,l110_jordan), []). fof(l112_jordan,axiom,( k22_euclid(k23_euclid(0,k1_real_1(3))) = k1_real_1(3) ), file(jordan,l112_jordan), []). fof(l151_jordan,axiom,( r2_hidden(k23_euclid(0,3),k1_sppol_2(k1_real_1(1),1,k1_real_1(3),3)) ), file(jordan,l151_jordan), []). fof(l152_jordan,axiom,( r2_hidden(k23_euclid(0,k1_real_1(3)),k1_sppol_2(k1_real_1(1),1,k1_real_1(3),3)) ), file(jordan,l152_jordan), []). fof(l158_jordan,axiom,( r1_tarski(k1_sppol_2(k1_real_1(1),1,k1_real_1(3),3),k2_jgraph_6(k1_real_1(1),1,k1_real_1(3),3)) ), file(jordan,l158_jordan), []). fof(rc10_pscomp_1,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) & v10_pscomp_1(A) ) ), file(pscomp_1,rc10_pscomp_1), []). fof(rc1_borsuk_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A)) & v5_pre_topc(B,A,A) & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ), file(borsuk_2,rc1_borsuk_2), []). fof(rc1_borsuk_4,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) & v2_connsp_1(B,A) & v6_compts_1(B,A) ) ) ), file(borsuk_4,rc1_borsuk_4), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_funct_2,axiom,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,A,B) ) ), file(funct_2,rc1_funct_2), []). fof(rc1_isomichi,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v2_tdlat_3(A) & ~ v3_realset2(A) ) ), file(isomichi,rc1_isomichi), []). fof(rc1_jordan,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) & ~ v1_xboole_0(B) & v3_pre_topc(B,k15_euclid(A)) & v4_pre_topc(B,k15_euclid(A)) & ~ v1_jordan2c(B,A) & v1_jordan1(B,A) ) ) ), file(jordan,rc1_jordan), []). fof(rc1_jordan1a,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & v4_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & v1_jordan2c(A,2) & v1_topreal2(A) ) ), file(jordan1a,rc1_jordan1a), []). fof(rc1_jordan21,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & v4_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & ~ v1_realset1(A) & v1_topreal2(A) & v1_jordan21(A) ) ), file(jordan21,rc1_jordan21), []). fof(rc1_jordan5a,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ? [B] : ( m1_pre_topc(B,k15_euclid(A)) & ~ v3_struct_0(B) & v1_pre_topc(B) & v2_pre_topc(B) & v2_compts_1(B) & v3_compts_1(B) ) ) ), file(jordan5a,rc1_jordan5a), []). 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_pscomp_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) ), file(pscomp_1,rc1_pscomp_1), []). fof(rc1_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(realset1,rc1_realset1), []). fof(rc1_seq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ), file(seq_1,rc1_seq_1), []). 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_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) & v2_tdlat_3(B) & v3_tdlat_3(B) & v4_tex_4(B,A) ) ) ), file(tex_4,rc1_tex_4), []). fof(rc1_topgen_3,axiom,( ! [A] : ( ~ v1_finset_1(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & ~ v1_finset_1(B) ) ) ), file(topgen_3,rc1_topgen_3), []). fof(rc1_topreal6,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v2_compts_1(A) & v6_group_1(A) & v10_pscomp_1(A) ) ), file(topreal6,rc1_topreal6), []). fof(rc1_topreala,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v1_monoid_0(A) & v2_monoid_0(A) ) ), file(topreala,rc1_topreala), []). fof(rc1_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) ) ) ), file(tops_1,rc1_tops_1), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_borsuk_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ? [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) & ~ v1_xboole_0(C) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(C,A,B) & v1_partfun1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ), file(borsuk_1,rc2_borsuk_1), []). fof(rc2_borsuk_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_borsuk_2(C,A,B,B) & ~ v1_xboole_0(C) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) & v5_pre_topc(C,k5_topmetr,A) & v1_partfun1(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) ) ), file(borsuk_2,rc2_borsuk_2), []). fof(rc2_borsuk_4,axiom,( ! [A] : ( ( ~ v3_realset2(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & ~ v1_realset1(B) ) ) ), file(borsuk_4,rc2_borsuk_4), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_jordan,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) & ~ v1_xboole_0(B) & v4_pre_topc(B,k15_euclid(A)) & v1_jordan2c(B,A) & v1_jordan1(B,A) ) ) ), file(jordan,rc2_jordan), []). fof(rc2_jordan1a,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & v4_pre_topc(A,k15_euclid(2)) & v6_compts_1(A,k15_euclid(2)) & v1_sppol_1(A) & v1_jordan2c(A,2) ) ), file(jordan1a,rc2_jordan1a), []). 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_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_pscomp_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_setfam_1(A) ) ), file(pscomp_1,rc2_pscomp_1), []). fof(rc2_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) ), file(realset1,rc2_realset1), []). 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_tex_4,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & ~ v2_tex_2(B,A) ) ) ), file(tex_4,rc2_tex_4), []). fof(rc2_topreal6,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & v2_connsp_1(A,k15_euclid(2)) & v1_jordan1(A,2) ) ), file(topreal6,rc2_topreal6), []). fof(rc2_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ), file(tops_1,rc2_tops_1), []). 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_borsuk_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) & v1_borsuk_1(B,A) ) ) ), file(borsuk_1,rc3_borsuk_1), []). fof(rc3_borsuk_2,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) ) ), file(borsuk_2,rc3_borsuk_2), []). fof(rc3_borsuk_4,axiom,( ! [A] : ( ~ v1_realset1(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & ~ v1_realset1(B) ) ) ), file(borsuk_4,rc3_borsuk_4), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_jordan,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) & ~ v1_xboole_0(B) & v3_pre_topc(B,k15_euclid(A)) & v1_jordan2c(B,A) & v1_jordan1(B,A) ) ) ), file(jordan,rc3_jordan), []). fof(rc3_jordan1a,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ~ v1_xboole_0(A) & v4_pre_topc(A,k15_euclid(2)) & v6_compts_1(A,k15_euclid(2)) & v2_sppol_1(A) & v1_jordan2c(A,2) ) ), file(jordan1a,rc3_jordan1a), []). 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_pscomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v1_setfam_1(A) ) ), file(pscomp_1,rc3_pscomp_1), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_tops_1,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) & v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ), file(tops_1,rc3_tops_1), []). 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_borsuk_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) & v2_pre_topc(B) & v1_borsuk_1(B,A) ) ) ), file(borsuk_1,rc4_borsuk_1), []). fof(rc4_isomichi,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v2_tdlat_3(A) & ~ v3_realset2(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v3_isomichi(B,A) & v4_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ), file(isomichi,rc4_isomichi), []). 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_pscomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & ~ v1_setfam_1(A) ) ), file(pscomp_1,rc4_pscomp_1), []). fof(rc4_rcomp_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v2_connsp_1(B,A) ) ) ), file(rcomp_3,rc4_rcomp_3), []). fof(rc4_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) ) ) ), file(tops_1,rc4_tops_1), []). fof(rc4_waybel_3,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) & v3_compts_1(A) & v4_compts_1(A) & v5_compts_1(A) & v6_waybel_3(A) ) ), file(waybel_3,rc4_waybel_3), []). 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(rc4_yellow13,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v1_tdlat_3(A) & v2_tdlat_3(A) & v3_tdlat_3(A) & v4_tdlat_3(A) & v5_tdlat_3(A) & v6_group_1(A) & v3_realset2(A) ) ), file(yellow13,rc4_yellow13), []). 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_rcomp_3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v2_connsp_1(B,A) ) ) ), file(rcomp_3,rc5_rcomp_3), []). 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(rc5_tops_1,axiom,( ! [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) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ), file(tops_1,rc5_tops_1), []). 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(rc6_topgrp_1,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v3_realset2(A) & v1_pre_topc(A) & v2_pre_topc(A) & v1_topgrp_1(A) ) ), file(topgrp_1,rc6_topgrp_1), []). fof(rc7_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_isomichi(B,A) & ~ v4_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ), file(isomichi,rc7_isomichi), []). 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(rc7_pscomp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_relset_1(B,u1_struct_0(A),k1_numbers) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & v1_seq_1(B) & v1_partfun1(B,u1_struct_0(A),k1_numbers) & v9_pscomp_1(B,A) ) ) ), file(pscomp_1,rc7_pscomp_1), []). fof(rc8_pscomp_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & m1_pre_topc(B,A) ) => ? [C] : ( m1_relset_1(C,u1_struct_0(B),k1_numbers) & ~ v1_xboole_0(C) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(B),k1_numbers) & v1_seq_1(C) & v1_partfun1(C,u1_struct_0(B),k1_numbers) & v9_pscomp_1(C,B) ) ) ), file(pscomp_1,rc8_pscomp_1), []). fof(rc9_pscomp_1,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) & v6_compts_1(B,A) ) ) ), file(pscomp_1,rc9_pscomp_1), []). fof(redefinition_k1_pscomp_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => k1_pscomp_1(A,B,C) = k2_relat_1(C) ) ), file(pscomp_1,k1_pscomp_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_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ), file(funct_2,k2_funct_2), []). fof(redefinition_k2_struct_0,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k2_struct_0(A,B,C) = k2_tarski(B,C) ) ), file(struct_0,k2_struct_0), []). fof(redefinition_k3_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k3_pscomp_1(A) = k2_seq_4(A) ) ), file(pscomp_1,k3_pscomp_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_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k4_pscomp_1(A) = k3_seq_4(A) ) ), file(pscomp_1,k4_pscomp_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_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_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_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_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(redefinition_r1_borsuk_6,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ( r1_borsuk_6(A,B,C) <=> r1_borsuk_2(A,B,C) ) ) ), file(borsuk_6,r1_borsuk_6), []). fof(redefinition_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r2_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r2_subset_1), []). fof(reflexivity_r1_borsuk_6,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => r1_borsuk_6(A,B,B) ) ), file(borsuk_6,r1_borsuk_6), []). 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(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(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(spc3_boole,axiom,( ~ v1_xboole_0(3) ), file(boole,spc3_boole), []). fof(spc3_numerals,axiom, ( v2_xreal_0(3) & m1_subset_1(3,k5_numbers) ), file(numerals,spc3_numerals), []). fof(symmetry_r1_borsuk_6,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ( r1_borsuk_6(A,B,C) => r1_borsuk_6(A,C,B) ) ) ), file(borsuk_6,r1_borsuk_6), []). fof(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). fof(symmetry_r2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r2_subset_1(A,B) => r2_subset_1(B,A) ) ) ), file(subset_1,r2_subset_1), []). fof(t16_jordan1a,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( r1_tarski(k9_jordan6(A),A) & r1_tarski(k8_jordan6(A),A) ) ) ), file(jordan1a,t16_jordan1a), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). 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(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). 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(t39_jordan,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m1_pre_topc(B,A) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(B)) => ! [F] : ( m1_subset_1(F,u1_struct_0(B)) => ! [G] : ( m1_borsuk_2(G,A,C,D) => ( ( C = E & D = F & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(A),G),u1_struct_0(B)) ) => ( r1_borsuk_6(B,E,F) & m1_borsuk_2(G,B,E,F) ) ) ) ) ) ) ) ) ) ), file(jordan,t39_jordan), []). 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_xboole_0,axiom,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ ( r2_hidden(C,A) & r2_hidden(C,B) ) ) & ~ ( ? [C] : ( r2_hidden(C,A) & r2_hidden(C,B) ) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t3_xboole_0), []). 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(t51_jordan,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ~ ( r1_topreal1(k15_euclid(A),B,C,D) & ! [E] : ( m1_borsuk_2(E,k15_euclid(A),B,C) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),D))) & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),D))) ) => ~ ( k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),D)),F) = D & E = F ) ) ) ) ) ) ) ) ), file(jordan,t51_jordan), []). fof(t52_jordan,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_borsuk_2(D,k15_euclid(A),B,C) & ? [E] : ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),k3_topreal1(A,B,C)))) & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),k3_topreal1(A,B,C)))) & k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),k3_topreal1(A,B,C))),E) = k3_topreal1(A,B,C) & D = E ) ) ) ) ) ), file(jordan,t52_jordan), []). fof(t53_topreala,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => u1_struct_0(k1_topreala(A,B,C,D)) = k2_jgraph_6(A,B,C,D) ) ) ) ) ), file(topreala,t53_topreala), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). 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_jgraph_8,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ! [E] : ( m1_subset_1(E,u1_struct_0(k1_topreala(A,B,C,D))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k1_topreala(A,B,C,D))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k1_topreala(A,B,C,D))) => ! [H] : ( m1_subset_1(H,u1_struct_0(k1_topreala(A,B,C,D))) => ! [I] : ( m1_borsuk_2(I,k1_topreala(A,B,C,D),E,F) => ! [J] : ( m1_borsuk_2(J,k1_topreala(A,B,C,D),H,G) => ! [K] : ( m1_subset_1(K,u1_struct_0(k15_euclid(2))) => ! [L] : ( m1_subset_1(L,u1_struct_0(k15_euclid(2))) => ! [M] : ( m1_subset_1(M,u1_struct_0(k15_euclid(2))) => ! [N] : ( m1_subset_1(N,u1_struct_0(k15_euclid(2))) => ~ ( k21_euclid(K) = A & k21_euclid(L) = B & k22_euclid(M) = C & k22_euclid(N) = D & E = K & F = L & G = M & H = N & ! [O] : ( m1_subset_1(O,u1_struct_0(k22_borsuk_1)) => ! [P] : ( m1_subset_1(P,u1_struct_0(k22_borsuk_1)) => k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k1_topreala(A,B,C,D)),I,O) != k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k1_topreala(A,B,C,D)),J,P) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(jgraph_8,t6_jgraph_8), []). 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(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(t81_jordan,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( r1_jordan24(2,A,k23_euclid(k1_real_1(1),0),k23_euclid(1,0)) => r1_tarski(A,k2_jgraph_6(k1_real_1(1),1,k1_real_1(3),3)) ) ) ), file(jordan,t81_jordan), []). fof(t89_jordan,axiom,( ! [A] : ( ( v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( r1_jordan24(2,A,k23_euclid(k1_real_1(1),0),k23_euclid(1,0)) => ( k30_pscomp_1(A) = k23_euclid(k1_real_1(1),0) & k31_pscomp_1(A) = k23_euclid(k1_real_1(1),0) ) ) ) ), file(jordan,t89_jordan), []). 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), []). fof(t90_jordan,axiom,( ! [A] : ( ( v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( r1_jordan24(2,A,k23_euclid(k1_real_1(1),0),k23_euclid(1,0)) => ( k35_pscomp_1(A) = k23_euclid(1,0) & k34_pscomp_1(A) = k23_euclid(1,0) ) ) ) ), file(jordan,t90_jordan), []).