% Mizar problem: t79_jordan,jordan,4058,60 fof(t79_jordan,conjecture,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ( ( C = B & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),C) ) => A = k2_tops_1(k15_euclid(2),B) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_metric_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_isomichi,cc10_membered,cc10_topreal6,cc10_yellow13,cc11_isomichi,cc11_membered,cc11_topreal6,cc11_yellow13,cc12_isomichi,cc12_membered,cc13_isomichi,cc13_membered,cc14_isomichi,cc14_membered,cc15_isomichi,cc15_membered,cc16_isomichi,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_borsuk_4,cc1_funct_1,cc1_funct_2,cc1_isomichi,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_toprealb,cc1_tops_1,cc1_xreal_0,cc1_yellow13,cc20_membered,cc2_funct_1,cc2_isomichi,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_isomichi,cc3_jordan,cc3_membered,cc3_nat_1,cc3_tex_4,cc3_topreal6,cc3_topreala,cc3_tops_1,cc3_xreal_0,cc3_yellow13,cc4_borsuk_2,cc4_isomichi,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_isomichi,cc5_jordan,cc5_tex_4,cc5_topreal6,cc5_tops_1,cc5_xreal_0,cc5_yellow13,cc6_funct_2,cc6_isomichi,cc6_membered,cc6_tex_4,cc6_topreal6,cc6_tops_1,cc6_xreal_0,cc6_yellow13,cc7_isomichi,cc7_tex_4,cc7_topreal6,cc7_waybel_3,cc7_xreal_0,cc7_yellow13,cc8_isomichi,cc8_tex_4,cc8_topreal6,cc8_xreal_0,cc8_yellow13,cc9_isomichi,cc9_membered,cc9_topreal6,cc9_yellow13,commutativity_k2_xboole_0,commutativity_k3_xboole_0,commutativity_k4_subset_1,commutativity_k5_subset_1,connectedness_r1_xreal_0,d10_xboole_0,d1_euclid,d1_funct_2,d1_xboole_0,d2_jordan2c,d2_tops_1,d2_xboole_0,d3_abian,d3_jordan2c,d3_tarski,d3_xboole_0,d4_tarski,d5_abian,d5_jordan2c,d7_euclid,d8_euclid,dt_g1_metric_1,dt_g1_pre_topc,dt_k10_jordan,dt_k13_euclid,dt_k14_euclid,dt_k15_euclid,dt_k1_connsp_1,dt_k1_euclid,dt_k1_funct_1,dt_k1_jordan2c,dt_k1_numbers,dt_k1_pre_topc,dt_k1_relat_1,dt_k1_struct_0,dt_k1_tarski,dt_k1_topreal9,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_euclid,dt_k2_brouwer,dt_k2_partfun1,dt_k2_topreal9,dt_k2_tops_1,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_pre_topc,dt_k3_subset_1,dt_k3_tarski,dt_k3_topreal9,dt_k3_xboole_0,dt_k4_finseq_2,dt_k4_relset_1,dt_k4_subset_1,dt_k4_xboole_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_relat_1,dt_k5_subset_1,dt_k5_toprns_1,dt_k6_partfun1,dt_k6_pre_topc,dt_k6_relat_1,dt_k6_subset_1,dt_k7_funct_2,dt_k7_relat_1,dt_k7_toprealb,dt_k8_funct_2,dt_k9_jordan,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,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_finseq_2,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_jordan,fc10_membered,fc10_pscomp_1,fc11_isomichi,fc11_jordan,fc11_membered,fc11_tops_1,fc12_jordan,fc12_tops_1,fc13_jordan,fc13_tops_1,fc14_jordan,fc14_topreal9,fc15_topreal9,fc19_toprealb,fc1_euclid,fc1_funct_1,fc1_gobrd14,fc1_jordan1k,fc1_jordan5a,fc1_pre_topc,fc1_pscomp_1,fc1_seq_1,fc1_struct_0,fc1_subset_1,fc1_tex_4,fc1_topreal6,fc1_topreal9,fc1_tops_1,fc1_yellow13,fc20_toprealb,fc21_toprealb,fc22_membered,fc22_toprealb,fc23_membered,fc24_membered,fc25_membered,fc25_toprealb,fc26_membered,fc27_membered,fc28_membered,fc29_membered,fc2_euclid,fc2_funct_1,fc2_membered,fc2_pscomp_1,fc2_seq_1,fc2_subset_1,fc2_tex_4,fc2_topalg_5,fc2_topreal9,fc2_tops_1,fc2_yellow13,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc37_membered,fc38_membered,fc39_membered,fc3_brouwer,fc3_isomichi,fc3_jordan,fc3_pre_topc,fc3_realset1,fc3_tex_4,fc3_topalg_5,fc3_topgen_3,fc3_topreal9,fc3_tops_1,fc40_membered,fc41_membered,fc4_brouwer,fc4_funct_1,fc4_isomichi,fc4_pre_topc,fc4_subset_1,fc4_tex_4,fc4_topgen_3,fc4_topreal9,fc4_tops_1,fc5_isomichi,fc5_membered,fc5_topreal9,fc6_membered,fc6_topreal9,fc7_membered,fc7_topreal9,fc7_tops_1,fc8_isomichi,fc8_membered,fc8_topgen_3,fc8_topreal9,fc9_membered,fc9_topreal9,fraenkel_a_1_0_jordan,fraenkel_a_2_0_jordan2c,fraenkel_a_2_6_jordan,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_k3_subset_1,irreflexivity_r1_subset_1,l101_jordan,l102_jordan,l103_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_jordan2c,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_toprealb,rc1_tops_1,rc1_xreal_0,rc2_borsuk_1,rc2_borsuk_4,rc2_funct_1,rc2_isomichi,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_isomichi,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_topgrp_1,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,rc9_pscomp_1,redefinition_k1_struct_0,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k5_subset_1,redefinition_k6_partfun1,redefinition_k6_subset_1,redefinition_k7_funct_2,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,redefinition_r1_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,spc0_boole,spc2_boole,spc2_numerals,symmetry_r1_subset_1,symmetry_r1_xboole_0,t14_brouwer,t16_jordan2c,t16_topreal9,t17_jordan2c,t17_topreal9,t17_tsep_1,t18_topreal9,t19_topreal9,t1_boole,t1_jordan1,t1_numerals,t1_real,t1_subset,t1_xboole_1,t21_connsp_2,t22_funct_1,t23_funct_1,t24_connsp_1,t25_jordan,t26_jordan,t26_subset_1,t29_toprns_1,t2_boole,t2_real,t2_subset,t2_tarski,t32_jordan,t34_connsp_1,t34_jordan,t35_funct_1,t37_connsp_1,t38_connsp_1,t3_boole,t3_brouwer,t3_real,t3_subset,t3_xboole_0,t40_connsp_1,t43_connsp_1,t48_pre_topc,t4_boole,t4_real,t4_subset,t4_topreal1,t4_tsep_1,t50_subset_1,t52_pre_topc,t58_tops_2,t5_real,t5_subset,t61_tops_1,t6_boole,t6_real,t72_funct_1,t73_jordan,t74_jordan,t75_jordan,t76_jordan,t76_topreal6,t77_jordan,t7_boole,t7_funct_2,t7_real,t7_topmetr,t7_topreal9,t7_xboole_1,t84_borsuk_4,t89_jordan2c,t8_boole,t8_real,t8_xboole_1,t9_funct_2,t9_topreal9,t9_toprealb]), [file(jordan,t79_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_isomichi,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) & v4_tops_1(B,A) ) => ( v4_pre_topc(B,A) & v5_tops_1(B,A) ) ) ) ) ), file(isomichi,cc10_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_isomichi(B,A) => ( v3_isomichi(B,A) & ~ v4_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ) ) ), file(isomichi,cc14_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_isomichi(B,A) => ( v3_isomichi(B,A) & ~ v4_isomichi(B,A) & ~ v5_isomichi(B,A) ) ) ) ) ), file(isomichi,cc15_isomichi), []). 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_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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v4_tops_1(B,A) => ( v1_isomichi(B,A) & v2_isomichi(B,A) ) ) ) ) ), file(isomichi,cc1_isomichi), []). 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_toprealb,axiom,( ! [A] : ( m1_pre_topc(A,k15_euclid(2)) => ( v1_toprealb(A) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) & v10_pscomp_1(A) & v1_borsuk_2(A) & v1_monoid_0(A) & v2_monoid_0(A) ) ) ) ), file(toprealb,cc1_toprealb), []). 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_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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v1_isomichi(B,A) & v2_isomichi(B,A) ) => v4_tops_1(B,A) ) ) ) ), file(isomichi,cc2_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_isomichi(B,A) => v1_decomp_1(B,A) ) ) ) ), file(isomichi,cc3_isomichi), []). 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_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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_decomp_1(B,A) => v2_isomichi(B,A) ) ) ) ), file(isomichi,cc4_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v6_tops_1(B,A) => v3_pre_topc(B,A) ) ) ) ), file(isomichi,cc5_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v5_tops_1(B,A) => v4_pre_topc(B,A) ) ) ) ), file(isomichi,cc6_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v6_tops_1(B,A) => ( v3_pre_topc(B,A) & v4_tops_1(B,A) & v1_decomp_1(B,A) & v1_isomichi(B,A) & v2_isomichi(B,A) ) ) ) ) ), file(isomichi,cc7_isomichi), []). 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_isomichi,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_tops_1(B,A) ) => ( v3_pre_topc(B,A) & v6_tops_1(B,A) ) ) ) ) ), file(isomichi,cc8_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v5_tops_1(B,A) => ( v4_pre_topc(B,A) & v4_tops_1(B,A) & v1_decomp_1(B,A) & v1_isomichi(B,A) & v2_isomichi(B,A) ) ) ) ) ), file(isomichi,cc9_isomichi), []). 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_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). 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(d10_xboole_0,axiom,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), []). 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_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d2_jordan2c,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_jordan2c(B,A) <=> ( v6_tbsp_1(B,k14_euclid(A)) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) ) ) ) ), file(jordan2c,d2_jordan2c), []). fof(d2_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => k2_tops_1(A,B) = k5_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,k3_subset_1(u1_struct_0(A),B))) ) ) ), file(tops_1,d2_tops_1), []). fof(d2_xboole_0,axiom,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), []). fof(d3_abian,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_abian(A,B) <=> ( r2_hidden(A,k1_relat_1(B)) & A = k1_funct_1(B,A) ) ) ) ), file(abian,d3_abian), []). fof(d3_jordan2c,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( r1_jordan2c(A,B,C) <=> ( r4_connsp_1(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B),C) & v1_jordan2c(C,A) ) ) ) ) ) ), file(jordan2c,d3_jordan2c), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d3_xboole_0,axiom,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), []). fof(d4_tarski,axiom,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), []). fof(d5_abian,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( r3_abian(A) <=> ? [B] : r1_abian(B,A) ) ) ), file(abian,d5_abian), []). fof(d5_jordan2c,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => k1_jordan2c(A,B) = k3_tarski(a_2_0_jordan2c(A,B)) ) ) ), file(jordan2c,d5_jordan2c), []). 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(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_jordan,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & 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(D) & v2_xreal_0(D) ) => ( v1_funct_1(k10_jordan(A,B,C,D)) & v1_funct_2(k10_jordan(A,B,C,D),u1_struct_0(k7_toprealb(A,B,D)),u1_struct_0(k7_toprealb(A,B,D))) & m2_relset_1(k10_jordan(A,B,C,D),u1_struct_0(k7_toprealb(A,B,D)),u1_struct_0(k7_toprealb(A,B,D))) ) ) ), file(jordan,k10_jordan), []). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), []). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), []). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), []). fof(dt_k1_connsp_1,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_connsp_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(connsp_1,k1_connsp_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_jordan2c,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => m1_subset_1(k1_jordan2c(A,B),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(jordan2c,k1_jordan2c), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k1_pre_topc), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(struct_0,k1_struct_0), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => m1_subset_1(k1_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal9,k1_topreal9), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k20_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k20_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k20_euclid), []). fof(dt_k2_brouwer,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => m1_pre_topc(k2_brouwer(A,B,C),k15_euclid(A)) ) ), file(brouwer,k2_brouwer), []). fof(dt_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => ( v1_funct_1(k2_partfun1(A,B,C,D)) & m2_relset_1(k2_partfun1(A,B,C,D),A,B) ) ) ), file(partfun1,k2_partfun1), []). fof(dt_k2_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => m1_subset_1(k2_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal9,k2_topreal9), []). fof(dt_k2_tops_1,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k2_tops_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(tops_1,k2_tops_1), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), []). fof(dt_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => m1_subset_1(k3_subset_1(A,B),k1_zfmisc_1(A)) ) ), file(subset_1,k3_subset_1), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k3_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => m1_subset_1(k3_topreal9(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal9,k3_topreal9), []). 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_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_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_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_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_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_toprns_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) ), file(toprns_1,k5_toprns_1), []). fof(dt_k6_partfun1,axiom,( ! [A] : ( v1_partfun1(k6_partfun1(A),A,A) & m2_relset_1(k6_partfun1(A),A,A) ) ), file(partfun1,k6_partfun1), []). fof(dt_k6_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k6_pre_topc), []). fof(dt_k6_relat_1,axiom,( ! [A] : v1_relat_1(k6_relat_1(A)) ), file(relat_1,k6_relat_1), []). fof(dt_k6_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k6_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k6_subset_1), []). fof(dt_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => ( v1_funct_1(k7_funct_2(A,B,C,D,E)) & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C) & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ), file(funct_2,k7_funct_2), []). 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_toprealb,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => m1_pre_topc(k7_toprealb(A,B,C),k15_euclid(A)) ) ), file(toprealb,k7_toprealb), []). 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_k9_jordan,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & 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(D) & v2_xreal_0(D) ) => ( v1_funct_1(k9_jordan(A,B,C,D)) & v1_funct_2(k9_jordan(A,B,C,D),u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),k2_topreal9(A,B,D),k1_struct_0(k15_euclid(A),C)))),u1_struct_0(k7_toprealb(A,B,D))) & m2_relset_1(k9_jordan(A,B,C,D),u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),k2_topreal9(A,B,D),k1_struct_0(k15_euclid(A),C)))),u1_struct_0(k7_toprealb(A,B,D))) ) ) ), file(jordan,k9_jordan), []). 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_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_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_jordan,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) & m1_subset_1(D,u1_struct_0(k15_euclid(B))) ) => ~ v1_xboole_0(k4_xboole_0(k2_topreal9(B,C,A),k1_struct_0(k15_euclid(B),D))) ) ), file(jordan,fc10_jordan), []). fof(fc10_membered,axiom,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), []). 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(fc11_isomichi,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v3_isomichi(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v3_isomichi(k3_subset_1(u1_struct_0(A),B),A) & ~ v4_isomichi(k3_subset_1(u1_struct_0(A),B),A) & ~ v5_isomichi(k3_subset_1(u1_struct_0(A),B),A) ) ) ), file(isomichi,fc11_isomichi), []). fof(fc11_jordan,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_jordan2c(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => ( v4_pre_topc(k6_pre_topc(k15_euclid(A),B),k15_euclid(A)) & v1_jordan2c(k6_pre_topc(k15_euclid(A),B),A) ) ) ), file(jordan,fc11_jordan), []). fof(fc11_membered,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), []). fof(fc11_tops_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v3_tops_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v1_tops_1(k3_subset_1(u1_struct_0(A),B),A) ) ), file(tops_1,fc11_tops_1), []). fof(fc12_jordan,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_jordan2c(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => ( v4_pre_topc(k2_tops_1(k15_euclid(A),B),k15_euclid(A)) & v1_jordan2c(k2_tops_1(k15_euclid(A),B),A) ) ) ), file(jordan,fc12_jordan), []). fof(fc12_tops_1,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))) ) => ( v2_tops_1(k2_tops_1(A,B),A) & v3_tops_1(k2_tops_1(A,B),A) ) ) ), file(tops_1,fc12_tops_1), []). fof(fc13_jordan,axiom,( ! [A,B] : ( ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & v1_xreal_0(B) ) => ( v2_pre_topc(k2_brouwer(2,A,B)) & v1_borsuk_1(k2_brouwer(2,A,B),k15_euclid(2)) & v1_topalg_2(k2_brouwer(2,A,B),2) ) ) ), file(jordan,fc13_jordan), []). fof(fc13_tops_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v4_pre_topc(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v2_tops_1(k2_tops_1(A,B),A) & v3_tops_1(k2_tops_1(A,B),A) ) ) ), file(tops_1,fc13_tops_1), []). fof(fc14_jordan,axiom,( ! [A,B] : ( ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & v1_xreal_0(B) ) => ( v2_pre_topc(k2_brouwer(2,A,B)) & v2_compts_1(k2_brouwer(2,A,B)) & v1_borsuk_1(k2_brouwer(2,A,B),k15_euclid(2)) & v1_topalg_2(k2_brouwer(2,A,B),2) ) ) ), file(jordan,fc14_jordan), []). fof(fc14_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & v3_xreal_0(C) ) => ( v1_xboole_0(k3_topreal9(A,B,C)) & v4_pre_topc(k3_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k3_topreal9(A,B,C),A) ) ) ), file(topreal9,fc14_topreal9), []). fof(fc15_topreal9,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & ~ v3_xreal_0(C) ) => ( ~ v1_xboole_0(k3_topreal9(A,B,C)) & v4_pre_topc(k3_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k3_topreal9(A,B,C),A) ) ) ), file(topreal9,fc15_topreal9), []). fof(fc19_toprealb,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & m1_subset_1(B,u1_struct_0(k15_euclid(2))) ) => ( ~ v1_xboole_0(k3_topreal9(2,B,A)) & v4_pre_topc(k3_topreal9(2,B,A),k15_euclid(2)) & v1_jordan2c(k3_topreal9(2,B,A),2) & v2_connsp_1(k3_topreal9(2,B,A),k15_euclid(2)) & v6_compts_1(k3_topreal9(2,B,A),k15_euclid(2)) & v1_topreal2(k3_topreal9(2,B,A)) ) ) ), file(toprealb,fc19_toprealb), []). 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_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k5_relat_1(A,B)) & v1_funct_1(k5_relat_1(A,B)) ) ) ), file(funct_1,fc1_funct_1), []). fof(fc1_gobrd14,axiom,( ! [A] : ( ( v4_pre_topc(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => v3_pre_topc(k1_jordan2c(2,A),k15_euclid(2)) ) ), file(gobrd14,fc1_gobrd14), []). fof(fc1_jordan1k,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( ~ v1_xboole_0(k1_jordan2c(2,A)) & v3_pre_topc(k1_jordan2c(2,A),k15_euclid(2)) ) ) ), file(jordan1k,fc1_jordan1k), []). 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_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => ( v1_xboole_0(k1_pre_topc(A)) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(pre_topc,fc1_pre_topc), []). 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_tex_4,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v1_xboole_0(k6_pre_topc(A,B)) & v4_pre_topc(k6_pre_topc(A,B),A) ) ) ), file(tex_4,fc1_tex_4), []). 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_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & ~ v2_xreal_0(C) ) => v1_xboole_0(k1_topreal9(A,B,C)) ) ), file(topreal9,fc1_topreal9), []). fof(fc1_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_xboole_0(k1_pre_topc(A)) & v4_pre_topc(k1_pre_topc(A),A) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(tops_1,fc1_tops_1), []). 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(fc20_toprealb,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & ~ v3_xreal_0(C) ) => ( ~ v3_struct_0(k7_toprealb(A,B,C)) & v1_pre_topc(k7_toprealb(A,B,C)) & v2_pre_topc(k7_toprealb(A,B,C)) & v1_monoid_0(k7_toprealb(A,B,C)) & v2_monoid_0(k7_toprealb(A,B,C)) ) ) ), file(toprealb,fc20_toprealb), []). fof(fc21_toprealb,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xboole_0(C) & v1_xreal_0(C) ) => ( v2_pre_topc(k7_toprealb(A,B,C)) & v3_realset2(k7_toprealb(A,B,C)) & v1_monoid_0(k7_toprealb(A,B,C)) & v2_monoid_0(k7_toprealb(A,B,C)) ) ) ), file(toprealb,fc21_toprealb), []). fof(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc22_toprealb,axiom,( ! [A,B] : ( ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & v1_xreal_0(B) & v2_xreal_0(B) ) => ( ~ v3_struct_0(k7_toprealb(2,A,B)) & v1_pre_topc(k7_toprealb(2,A,B)) & v2_pre_topc(k7_toprealb(2,A,B)) & v2_compts_1(k7_toprealb(2,A,B)) & v10_pscomp_1(k7_toprealb(2,A,B)) & v1_borsuk_2(k7_toprealb(2,A,B)) & v1_monoid_0(k7_toprealb(2,A,B)) & v2_monoid_0(k7_toprealb(2,A,B)) & v1_toprealb(k7_toprealb(2,A,B)) ) ) ), file(toprealb,fc22_toprealb), []). 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(fc25_toprealb,axiom,( ! [A,B] : ( ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v3_struct_0(k7_toprealb(2,A,B)) & v1_pre_topc(k7_toprealb(2,A,B)) & v2_pre_topc(k7_toprealb(2,A,B)) & v1_borsuk_2(k7_toprealb(2,A,B)) & v1_monoid_0(k7_toprealb(2,A,B)) & v2_monoid_0(k7_toprealb(2,A,B)) ) ) ), file(toprealb,fc25_toprealb), []). 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(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). 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(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_funct_1,axiom,( ! [A] : ( v1_relat_1(k6_relat_1(A)) & v1_funct_1(k6_relat_1(A)) ) ), file(funct_1,fc2_funct_1), []). 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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_tex_4,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_xboole_0(k6_pre_topc(A,B)) & v3_pre_topc(k6_pre_topc(A,B),A) & v4_pre_topc(k6_pre_topc(A,B),A) & v2_tops_1(k6_pre_topc(A,B),A) & v3_tops_1(k6_pre_topc(A,B),A) & v1_tex_2(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A))) & v1_membered(k6_pre_topc(A,B)) & v2_membered(k6_pre_topc(A,B)) & v3_membered(k6_pre_topc(A,B)) & v4_membered(k6_pre_topc(A,B)) & v5_membered(k6_pre_topc(A,B)) ) ) ), file(tex_4,fc2_tex_4), []). 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_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & v2_xreal_0(C) ) => ~ v1_xboole_0(k1_topreal9(A,B,C)) ) ), file(topreal9,fc2_topreal9), []). fof(fc2_tops_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v4_pre_topc(k6_pre_topc(A,B),A) ) ), file(tops_1,fc2_tops_1), []). 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(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(fc37_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k4_xboole_0(A,B)) ) ), file(membered,fc37_membered), []). fof(fc38_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc38_membered), []). fof(fc39_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc39_membered), []). fof(fc3_brouwer,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & ~ v3_xreal_0(C) ) => ( ~ v3_struct_0(k2_brouwer(A,B,C)) & v2_pre_topc(k2_brouwer(A,B,C)) ) ) ), file(brouwer,fc3_brouwer), []). fof(fc3_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_xboole_0(k1_pre_topc(A)) & v3_pre_topc(k1_pre_topc(A),A) & v4_pre_topc(k1_pre_topc(A),A) & v2_tops_1(k1_pre_topc(A),A) & v3_tops_1(k1_pre_topc(A),A) & v5_tops_1(k1_pre_topc(A),A) & v6_tops_1(k1_pre_topc(A),A) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(isomichi,fc3_isomichi), []). fof(fc3_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(k5_toprns_1(A,k20_euclid(A,B,C))) & ~ v3_xreal_0(k5_toprns_1(A,k20_euclid(A,B,C))) & v1_xcmplx_0(k5_toprns_1(A,k20_euclid(A,B,C))) ) ) ), file(jordan,fc3_jordan), []). 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_realset1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) & v1_realset1(k1_tarski(A)) ) ), file(realset1,fc3_realset1), []). fof(fc3_tex_4,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v1_tex_2(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v1_xboole_0(k6_pre_topc(A,B)) & v4_pre_topc(k6_pre_topc(A,B),A) & ~ v1_tex_2(k6_pre_topc(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ), file(tex_4,fc3_tex_4), []). 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_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & v3_xreal_0(C) ) => v1_xboole_0(k2_topreal9(A,B,C)) ) ), file(topreal9,fc3_topreal9), []). fof(fc3_tops_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v4_pre_topc(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ), file(tops_1,fc3_tops_1), []). fof(fc40_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc40_membered), []). fof(fc41_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) & v5_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc41_membered), []). fof(fc4_brouwer,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => ( v2_pre_topc(k2_brouwer(A,B,C)) & v1_topalg_2(k2_brouwer(A,B,C),A) ) ) ), file(brouwer,fc4_brouwer), []). 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_isomichi,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v6_tops_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v5_tops_1(k3_subset_1(u1_struct_0(A),B),A) ) ), file(isomichi,fc4_isomichi), []). 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_tex_4,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & ~ v3_realset2(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & ~ v1_realset1(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v1_xboole_0(k6_pre_topc(A,B)) & ~ v1_realset1(k6_pre_topc(A,B)) & v4_pre_topc(k6_pre_topc(A,B),A) ) ) ), file(tex_4,fc4_tex_4), []). 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(fc4_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) & ~ v3_xreal_0(C) ) => ~ v1_xboole_0(k2_topreal9(A,B,C)) ) ), file(topreal9,fc4_topreal9), []). fof(fc4_tops_1,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))) ) => v4_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ), file(tops_1,fc4_tops_1), []). fof(fc5_isomichi,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v5_tops_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v6_tops_1(k3_subset_1(u1_struct_0(A),B),A) ) ), file(isomichi,fc5_isomichi), []). 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_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => ( v3_pre_topc(k1_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k1_topreal9(A,B,C),A) ) ) ), file(topreal9,fc5_topreal9), []). 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_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => ( v4_pre_topc(k2_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k2_topreal9(A,B,C),A) ) ) ), file(topreal9,fc6_topreal9), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc7_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => ( v4_pre_topc(k3_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k3_topreal9(A,B,C),A) ) ) ), file(topreal9,fc7_topreal9), []). fof(fc7_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_xboole_0(k1_pre_topc(A)) & v3_pre_topc(k1_pre_topc(A),A) & v4_pre_topc(k1_pre_topc(A),A) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(tops_1,fc7_tops_1), []). fof(fc8_isomichi,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v4_pre_topc(k2_tops_1(A,B),A) ) ), file(isomichi,fc8_isomichi), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc8_topgen_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_tex_2(B,k1_zfmisc_1(A)) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ~ v1_xboole_0(k3_subset_1(A,B)) ) ), file(topgen_3,fc8_topgen_3), []). fof(fc8_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => ( v3_pre_topc(k1_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k1_topreal9(A,B,C),A) & v1_jordan1(k1_topreal9(A,B,C),A) ) ) ), file(topreal9,fc8_topreal9), []). fof(fc9_membered,axiom,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), []). fof(fc9_topreal9,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & v1_xreal_0(C) ) => ( v4_pre_topc(k2_topreal9(A,B,C),k15_euclid(A)) & v1_jordan2c(k2_topreal9(A,B,C),A) & v1_jordan1(k2_topreal9(A,B,C),A) ) ) ), file(topreal9,fc9_topreal9), []). fof(fraenkel_a_1_0_jordan,axiom,( ! [A,B] : ( ( v1_topreal2(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( r2_hidden(A,a_1_0_jordan(B)) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & A = C & r1_jordan2c(2,B,C) ) ) ) ), file(jordan,a_1_0_jordan), []). fof(fraenkel_a_2_0_jordan2c,axiom,( ! [A,B,C] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) ) => ( r2_hidden(A,a_2_0_jordan2c(B,C)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) & A = D & r1_jordan2c(B,C,D) ) ) ) ), file(jordan2c,a_2_0_jordan2c), []). fof(fraenkel_a_2_6_jordan,axiom,( ! [A,B,C] : ( ( v1_topreal2(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B))))) ) => ( r2_hidden(A,a_2_6_jordan(B,C)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B))))) & A = D & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B)),D) & D != C ) ) ) ), file(jordan,a_2_6_jordan), []). 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_k3_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,k3_subset_1(A,B)) = B ) ), file(subset_1,k3_subset_1), []). fof(irreflexivity_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ r1_subset_1(A,A) ) ), file(subset_1,r1_subset_1), []). fof(l101_jordan,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [E] : ( ( v1_xreal_0(E) & ~ v3_xreal_0(E) ) => ~ ( r1_topreal1(k15_euclid(2),A,B,D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k2_brouwer(2,C,E)))) & ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k2_brouwer(2,C,E)),u1_struct_0(k3_pre_topc(k15_euclid(2),D))) & m2_relset_1(F,u1_struct_0(k2_brouwer(2,C,E)),u1_struct_0(k3_pre_topc(k15_euclid(2),D))) ) => ~ ( v5_pre_topc(F,k2_brouwer(2,C,E),k3_pre_topc(k15_euclid(2),D)) & k2_partfun1(u1_struct_0(k2_brouwer(2,C,E)),u1_struct_0(k3_pre_topc(k15_euclid(2),D)),F,D) = k6_partfun1(D) ) ) ) ) ) ) ) ) ), file(jordan,l101_jordan), []). fof(l102_jordan,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( ( v1_topreal2(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),D))))) => ! [I] : ( ( v1_xreal_0(I) & v2_xreal_0(I) ) => ( ( r1_topreal1(k15_euclid(2),A,B,E) & r1_tarski(E,D) & r1_tarski(D,k1_topreal9(2,C,I)) & r2_hidden(C,H) & r1_tarski(k5_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),F),k3_subset_1(u1_struct_0(k15_euclid(2)),F)),E) & r1_tarski(F,k1_topreal9(2,C,I)) ) => ! [J] : ( ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),E))) & m2_relset_1(J,u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),E))) ) => ~ ( v5_pre_topc(J,k2_brouwer(2,C,I),k3_pre_topc(k15_euclid(2),E)) & k2_partfun1(u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),E)),J,E) = k6_partfun1(E) & H = F & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),D)),H) & G = k6_subset_1(u1_struct_0(k15_euclid(2)),k2_topreal9(2,C,I),k1_struct_0(k15_euclid(2),C)) & ! [K] : ( ( v1_funct_1(K) & v1_funct_2(K,u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G))) & m2_relset_1(K,u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G))) ) => ~ ( v5_pre_topc(K,k2_brouwer(2,C,I),k3_pre_topc(k15_euclid(2),G)) & ! [L] : ( m1_subset_1(L,u1_struct_0(k2_brouwer(2,C,I))) => ( ( r2_hidden(L,k6_pre_topc(k15_euclid(2),F)) => k8_funct_2(u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G)),K,L) = k8_funct_2(u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),E)),J,L) ) & ( r2_hidden(L,k3_subset_1(u1_struct_0(k15_euclid(2)),F)) => k8_funct_2(u1_struct_0(k2_brouwer(2,C,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G)),K,L) = L ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(jordan,l102_jordan), []). fof(l103_jordan,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( ( v1_topreal2(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B))))) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B))))) => ! [G] : ( ( ~ v1_xboole_0(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( E != F => ! [I] : ( ( v1_xreal_0(I) & v2_xreal_0(I) ) => ( ( r1_tarski(G,B) & r1_tarski(B,k1_topreal9(2,A,I)) & r2_hidden(A,F) & r1_tarski(k5_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),C),k3_subset_1(u1_struct_0(k15_euclid(2)),C)),G) ) => ( r1_xboole_0(k1_topreal9(2,A,I),C) | ! [J] : ( ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G))) & m2_relset_1(J,u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G))) ) => ~ ( v5_pre_topc(J,k2_brouwer(2,A,I),k3_pre_topc(k15_euclid(2),G)) & k2_partfun1(u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G)),J,G) = k6_partfun1(G) & E = C & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B)),E) & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),B)),F) & D = k6_subset_1(u1_struct_0(k15_euclid(2)),k2_topreal9(2,A,I),k1_struct_0(k15_euclid(2),A)) & ! [K] : ( ( v1_funct_1(K) & v1_funct_2(K,u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),D))) & m2_relset_1(K,u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),D))) ) => ~ ( v5_pre_topc(K,k2_brouwer(2,A,I),k3_pre_topc(k15_euclid(2),D)) & ! [L] : ( m1_subset_1(L,u1_struct_0(k2_brouwer(2,A,I))) => ( ( r2_hidden(L,k6_pre_topc(k15_euclid(2),C)) => k8_funct_2(u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),D)),K,L) = L ) & ( r2_hidden(L,k3_subset_1(u1_struct_0(k15_euclid(2)),C)) => k8_funct_2(u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),D)),K,L) = k8_funct_2(u1_struct_0(k2_brouwer(2,A,I)),u1_struct_0(k3_pre_topc(k15_euclid(2),G)),J,L) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(jordan,l103_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_jordan2c,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v6_tbsp_1(B,A) ) ) ), file(jordan2c,rc1_jordan2c), []). 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_toprealb,axiom,( ? [A] : ( m1_pre_topc(A,k15_euclid(2)) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v2_compts_1(A) & v10_pscomp_1(A) & v1_borsuk_2(A) & v1_monoid_0(A) & v2_monoid_0(A) & v1_toprealb(A) ) ), file(toprealb,rc1_toprealb), []). 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_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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v4_tops_1(B,A) & v1_isomichi(B,A) & v2_isomichi(B,A) ) ) ), file(isomichi,rc2_isomichi), []). 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_isomichi,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v5_tops_1(B,A) & v6_tops_1(B,A) ) ) ), file(isomichi,rc3_isomichi), []). 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_topgrp_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) & v1_tops_1(B,A) ) ) ), file(topgrp_1,rc4_topgrp_1), []). 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(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_struct_0,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & m1_subset_1(B,u1_struct_0(A)) ) => k1_struct_0(A,B) = k1_tarski(B) ) ), file(struct_0,k1_struct_0), []). fof(redefinition_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_partfun1(A,B,C,D) = k7_relat_1(C,D) ) ), file(partfun1,k2_partfun1), []). 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_k6_partfun1,axiom,( ! [A] : k6_partfun1(A) = k6_relat_1(A) ), file(partfun1,k6_partfun1), []). fof(redefinition_k6_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k6_subset_1(A,B,C) = k4_xboole_0(B,C) ) ), file(subset_1,k6_subset_1), []). fof(redefinition_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ), file(funct_2,k7_funct_2), []). 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_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) <=> r1_xboole_0(A,B) ) ) ), file(subset_1,r1_subset_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). 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(symmetry_r1_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ( r1_subset_1(A,B) => r1_subset_1(B,A) ) ) ), file(subset_1,r1_subset_1), []). 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(t14_brouwer,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k2_brouwer(2,B,A)),u1_struct_0(k2_brouwer(2,B,A))) & v5_pre_topc(C,k2_brouwer(2,B,A),k2_brouwer(2,B,A)) & m2_relset_1(C,u1_struct_0(k2_brouwer(2,B,A)),u1_struct_0(k2_brouwer(2,B,A))) ) => r3_abian(C) ) ) ) ), file(brouwer,t14_brouwer), []). fof(t16_jordan2c,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( v1_jordan2c(C,A) & r1_tarski(B,C) ) => v1_jordan2c(B,A) ) ) ) ) ), file(jordan2c,t16_jordan2c), []). fof(t16_topreal9,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => r1_tarski(k1_topreal9(A,C,B),k2_topreal9(A,C,B)) ) ) ) ), file(topreal9,t16_topreal9), []). fof(t17_jordan2c,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( r1_jordan2c(A,B,C) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B))))) & D = C & r3_connsp_1(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B)),D) & v6_tbsp_1(D,k14_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) ) ) ) ) ), file(jordan2c,t17_jordan2c), []). fof(t17_topreal9,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => r1_tarski(k3_topreal9(A,C,B),k2_topreal9(A,C,B)) ) ) ) ), file(topreal9,t17_topreal9), []). fof(t17_tsep_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_tsep_1(B,A) & m1_pre_topc(B,A) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( C = D => ( v3_pre_topc(D,B) <=> v3_pre_topc(C,A) ) ) ) ) ) ) ), file(tsep_1,t17_tsep_1), []). fof(t18_topreal9,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k4_subset_1(u1_struct_0(k15_euclid(A)),k1_topreal9(A,C,B),k3_topreal9(A,C,B)) = k2_topreal9(A,C,B) ) ) ) ), file(topreal9,t18_topreal9), []). fof(t19_topreal9,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => r1_xboole_0(k1_topreal9(A,C,B),k3_topreal9(A,C,B)) ) ) ) ), file(topreal9,t19_topreal9), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_jordan1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => u1_struct_0(k3_pre_topc(A,B)) = B ) ) ), file(jordan1,t1_jordan1), []). 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(t21_connsp_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_connsp_2(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r3_connsp_1(A,B) => v3_pre_topc(B,A) ) ) ) ) ), file(connsp_2,t21_connsp_2), []). fof(t22_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B))) => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ), file(funct_1,t22_funct_1), []). fof(t23_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ), file(funct_1,t23_funct_1), []). fof(t24_connsp_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( C = D => ( v2_connsp_1(C,A) <=> v2_connsp_1(D,B) ) ) ) ) ) ) ), file(connsp_1,t24_connsp_1), []). fof(t25_jordan,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => m1_subset_1(C,u1_struct_0(k2_brouwer(A,C,B))) ) ) ) ), file(jordan,t25_jordan), []). fof(t26_jordan,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(C))) => ( r1_xreal_0(A,B) => r1_tarski(k1_topreal9(C,D,A),k1_topreal9(C,D,B)) ) ) ) ) ) ), file(jordan,t26_jordan), []). fof(t26_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => r1_xboole_0(B,k3_subset_1(A,B)) ) ), file(subset_1,t26_subset_1), []). fof(t29_toprns_1,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))) => ( k5_toprns_1(A,k20_euclid(A,B,C)) = 0 <=> B = C ) ) ) ) ), file(toprns_1,t29_toprns_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(t32_jordan,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( ( ~ v1_xboole_0(C) & v1_xreal_0(C) ) => k2_tops_1(k15_euclid(A),k1_topreal9(A,B,C)) = k3_topreal9(A,B,C) ) ) ) ), file(jordan,t32_jordan), []). fof(t34_connsp_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))) => ~ ( r3_connsp_1(A,B) & B = k1_pre_topc(A) ) ) ) ), file(connsp_1,t34_connsp_1), []). fof(t34_jordan,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_jordan2c(B,A) & 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] : ( v1_xreal_0(D) & v2_xreal_0(D) & r1_tarski(B,k1_topreal9(A,C,D)) ) ) ) ) ), file(jordan,t34_jordan), []). fof(t35_funct_1,axiom,( ! [A,B] : ( r2_hidden(B,A) => k1_funct_1(k6_relat_1(A),B) = B ) ), file(funct_1,t35_funct_1), []). fof(t37_connsp_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r3_connsp_1(A,B) & r3_connsp_1(A,C) & B != C & ~ r1_xboole_0(B,C) ) ) ) ) ), file(connsp_1,t37_connsp_1), []). fof(t38_connsp_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_connsp_1(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r3_connsp_1(A,C) & ~ r1_xboole_0(B,C) & ~ r1_tarski(B,C) ) ) ) ) ) ), file(connsp_1,t38_connsp_1), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). fof(t3_brouwer,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => u1_struct_0(k2_brouwer(A,C,B)) = k2_topreal9(A,C,B) ) ) ) ), file(brouwer,t3_brouwer), []). 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(t40_connsp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => r2_hidden(B,k1_connsp_1(A,B)) ) ) ), file(connsp_1,t40_connsp_1), []). fof(t43_connsp_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))) => ( r3_connsp_1(A,B) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(A)) & B = k1_connsp_1(A,C) ) ) ) ) ), file(connsp_1,t43_connsp_1), []). fof(t48_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => r1_tarski(B,k6_pre_topc(A,B)) ) ) ), file(pre_topc,t48_pre_topc), []). fof(t4_boole,axiom,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), []). 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(t4_topreal1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( r1_topreal1(A,C,D,B) => ( r2_hidden(C,B) & r2_hidden(D,B) ) ) ) ) ) ) ), file(topreal1,t4_topreal1), []). fof(t4_tsep_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_pre_topc(C,A) => ( r1_tarski(u1_struct_0(B),u1_struct_0(C)) <=> m1_pre_topc(B,C) ) ) ) ) ), file(tsep_1,t4_tsep_1), []). fof(t50_subset_1,axiom,( ! [A] : ( A != k1_xboole_0 => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ! [C] : ( m1_subset_1(C,A) => ( ~ r2_hidden(C,B) => r2_hidden(C,k3_subset_1(A,B)) ) ) ) ) ), file(subset_1,t50_subset_1), []). fof(t52_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v4_pre_topc(B,A) => k6_pre_topc(A,B) = B ) & ( ( v2_pre_topc(A) & k6_pre_topc(A,B) = B ) => v4_pre_topc(B,A) ) ) ) ) ), file(pre_topc,t52_pre_topc), []). fof(t58_tops_2,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( ( ~ v3_struct_0(C) & l1_pre_topc(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(C),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) ) => ( ( v5_pre_topc(D,A,C) & v5_pre_topc(E,C,B) ) => v5_pre_topc(k7_funct_2(u1_struct_0(A),u1_struct_0(C),u1_struct_0(B),D,E),A,B) ) ) ) ) ) ) ), file(tops_2,t58_tops_2), []). 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(t61_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))) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r2_hidden(C,k2_tops_1(A,B)) <=> ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( ( v3_pre_topc(D,A) & r2_hidden(C,D) ) => ( ~ r1_xboole_0(B,D) & ~ r1_xboole_0(k3_subset_1(u1_struct_0(A),B),D) ) ) ) ) ) ) ) ), file(tops_1,t61_tops_1), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t72_funct_1,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,A) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t72_funct_1), []). fof(t73_jordan,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( ( v1_xreal_0(C) & v2_xreal_0(C) ) => ( m1_subset_1(B,u1_struct_0(k2_brouwer(2,A,C))) => v5_pre_topc(k9_jordan(2,A,B,C),k3_pre_topc(k15_euclid(2),k6_subset_1(u1_struct_0(k15_euclid(2)),k2_topreal9(2,A,C),k1_struct_0(k15_euclid(2),B))),k7_toprealb(2,A,C)) ) ) ) ) ), file(jordan,t73_jordan), []). fof(t74_jordan,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( ( v1_xreal_0(D) & v2_xreal_0(D) ) => ( r2_hidden(C,k1_topreal9(A,B,D)) => k2_partfun1(u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),k2_topreal9(A,B,D),k1_struct_0(k15_euclid(A),C)))),u1_struct_0(k7_toprealb(A,B,D)),k9_jordan(A,B,C,D),k3_topreal9(A,B,D)) = k6_partfun1(k3_topreal9(A,B,D)) ) ) ) ) ) ), file(jordan,t74_jordan), []). fof(t75_jordan,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( ( v1_xreal_0(C) & v2_xreal_0(C) ) => ( r2_hidden(B,k1_topreal9(2,A,C)) => v5_pre_topc(k10_jordan(2,A,B,C),k7_toprealb(2,A,C),k7_toprealb(2,A,C)) ) ) ) ) ), file(jordan,t75_jordan), []). fof(t76_jordan,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( ( v1_xreal_0(D) & v2_xreal_0(D) ) => ~ ( r2_hidden(C,k1_topreal9(A,B,D)) & r3_abian(k10_jordan(A,B,C,D)) ) ) ) ) ) ), file(jordan,t76_jordan), []). fof(t76_topreal6,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( v1_jordan2c(B,A) & v1_jordan2c(C,A) ) => v1_jordan2c(k4_subset_1(u1_struct_0(k15_euclid(A)),B,C),A) ) ) ) ) ), file(topreal6,t76_topreal6), []). fof(t77_jordan,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ( ( C = B & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),C) & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),D) ) => ( C = D | r1_xboole_0(k6_pre_topc(k15_euclid(2),B),D) ) ) ) ) ) ) ), file(jordan,t77_jordan), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | r2_hidden(k1_funct_1(D,C),B) ) ) ) ), file(funct_2,t7_funct_2), []). 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(t7_topmetr,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) ) => ( ( v5_pre_topc(D,A,C) & m1_pre_topc(C,B) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) ) => ( E = D => v5_pre_topc(E,A,B) ) ) ) ) ) ) ) ), file(topmetr,t7_topmetr), []). fof(t7_topreal9,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [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,k1_topreal9(A,D,B)) <=> ~ r1_xreal_0(B,k5_toprns_1(A,k20_euclid(A,C,D))) ) ) ) ) ) ), file(topreal9,t7_topreal9), []). fof(t7_xboole_1,axiom,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), []). fof(t84_borsuk_4,axiom,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( ( v4_pre_topc(B,k15_euclid(2)) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( r1_tarski(B,A) & B != A & ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ~ ( r1_topreal1(k15_euclid(2),C,D,E) & r1_tarski(B,E) & r1_tarski(E,A) ) ) ) ) ) ) ) ), file(borsuk_4,t84_borsuk_4), []). fof(t89_jordan2c,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => ( v3_pre_topc(B,k15_euclid(A)) => v1_connsp_2(k3_pre_topc(k15_euclid(A),B)) ) ) ) ), file(jordan2c,t89_jordan2c), []). 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(t8_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), []). fof(t9_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r1_tarski(B,C) => ( ( B = k1_xboole_0 & A != k1_xboole_0 ) | ( v1_funct_1(D) & v1_funct_2(D,A,C) & m2_relset_1(D,A,C) ) ) ) ) ), file(funct_2,t9_funct_2), []). fof(t9_topreal9,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [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,k3_topreal9(A,D,B)) <=> k5_toprns_1(A,k20_euclid(A,C,D)) = B ) ) ) ) ) ), file(topreal9,t9_topreal9), []). fof(t9_toprealb,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => u1_struct_0(k7_toprealb(A,C,B)) = k3_topreal9(A,C,B) ) ) ) ), file(toprealb,t9_toprealb), []).