% Mizar problem: t21_topalg_5,topalg_5,1063,26 fof(t21_topalg_5,conjecture,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k8_toprealb(2))))) => ( ( r1_pre_topc(k8_toprealb(2),A) & v1_tops_2(A,k8_toprealb(2)) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k6_borsuk_1(B,k5_topmetr)),u1_struct_0(k8_toprealb(2))) & v5_pre_topc(C,k6_borsuk_1(B,k5_topmetr),k8_toprealb(2)) & m2_relset_1(C,u1_struct_0(k6_borsuk_1(B,k5_topmetr)),u1_struct_0(k8_toprealb(2))) ) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => ? [E] : ( ~ v1_xboole_0(E) & m2_finseq_1(E,k1_numbers) & k2_seq_1(k5_numbers,k1_numbers,E,1) = 0 & k2_seq_1(k5_numbers,k1_numbers,E,k3_finseq_1(E)) = 1 & v1_seqm_3(E) & ? [F] : ( v3_pre_topc(F,B) & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B))) & r2_hidden(D,F) & ! [G] : ( v4_ordinal2(G) => ~ ( r2_hidden(G,k4_finseq_1(E)) & r2_hidden(k2_xcmplx_0(G,1),k4_finseq_1(E)) & ! [H] : ( ( ~ v1_xboole_0(H) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k8_toprealb(2)))) ) => ~ ( r2_hidden(H,A) & r1_tarski(k4_pre_topc(k6_borsuk_1(B,k5_topmetr),k8_toprealb(2),C,k2_zfmisc_1(F,k1_rcomp_1(k2_seq_1(k5_numbers,k1_numbers,E,G),k2_seq_1(k5_numbers,k1_numbers,E,k2_xcmplx_0(G,1))))),H) ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc10_yellow13,cc11_membered,cc11_topreal6,cc11_yellow13,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_borsuk_2,cc1_funct_1,cc1_membered,cc1_nat_1,cc1_pre_topc,cc1_realset1,cc1_relset_1,cc1_topalg_3,cc1_topreal6,cc1_toprealb,cc1_tops_1,cc1_xreal_0,cc1_yellow13,cc20_membered,cc2_funct_1,cc2_int_1,cc2_membered,cc2_nat_1,cc2_topalg_3,cc2_topreal6,cc2_tops_1,cc2_xreal_0,cc2_yellow13,cc3_borsuk_2,cc3_borsuk_5,cc3_int_1,cc3_membered,cc3_nat_1,cc3_topalg_2,cc3_tops_1,cc3_xreal_0,cc3_yellow13,cc4_borsuk_2,cc4_int_1,cc4_membered,cc4_topalg_2,cc4_topreal6,cc4_tops_1,cc4_xreal_0,cc4_yellow13,cc5_topreal6,cc5_tops_1,cc5_xreal_0,cc5_yellow13,cc6_membered,cc6_topreal6,cc6_tops_1,cc6_xreal_0,cc6_yellow13,cc7_topreal6,cc7_xreal_0,cc7_yellow13,cc8_topreal6,cc8_xreal_0,cc8_yellow13,cc9_membered,cc9_topreal6,cc9_yellow13,commutativity_k1_nat_1,commutativity_k2_tarski,commutativity_k2_xcmplx_0,connectedness_r1_xreal_0,d10_xboole_0,d1_compts_1,d1_finset_1,d1_funct_2,d1_rcomp_3,d1_setfam_1,d1_tarski,d1_tops_2,d2_rcomp_3,d2_zfmisc_1,d3_pre_topc,d3_rcomp_3,d3_tarski,d4_tarski,d5_borsuk_1,d5_funct_1,d5_tarski,d7_compts_1,d7_connsp_3,d8_pre_topc,dt_g1_pre_topc,dt_k12_mcart_1,dt_k13_binarith,dt_k15_euclid,dt_k1_card_1,dt_k1_connsp_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_pcomps_1,dt_k1_rcomp_1,dt_k1_rcomp_2,dt_k1_relat_1,dt_k1_setfam_1,dt_k1_struct_0,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k22_borsuk_1,dt_k23_borsuk_1,dt_k2_pre_topc,dt_k2_rcomp_1,dt_k2_rcomp_2,dt_k2_relat_1,dt_k2_seq_1,dt_k2_seq_4,dt_k2_tarski,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_finseq_1,dt_k3_pre_topc,dt_k3_pscomp_1,dt_k3_seq_4,dt_k3_tarski,dt_k3_topmetr,dt_k4_finseq_1,dt_k4_finseq_2,dt_k4_finseq_4,dt_k4_pre_topc,dt_k4_pscomp_1,dt_k4_relset_1,dt_k4_tarski,dt_k4_topmetr,dt_k5_finseq_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relset_1,dt_k5_setfam_1,dt_k5_topmetr,dt_k6_borsuk_1,dt_k6_connsp_3,dt_k6_setfam_1,dt_k7_borsuk_1,dt_k8_borsuk_1,dt_k8_funct_2,dt_k8_toprealb,dt_k9_relat_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m1_pre_topc,dt_m1_rcomp_3,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_rcomp_3,dt_m2_relset_1,dt_m2_subset_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_1,existence_m1_finseq_2,existence_m1_pre_topc,existence_m1_rcomp_3,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_rcomp_3,existence_m2_relset_1,existence_m2_subset_1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_xreal_0,fc12_membered,fc12_xreal_0,fc13_membered,fc14_membered,fc15_membered,fc16_membered,fc17_rcomp_3,fc18_borsuk_5,fc1_borsuk_5,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_topalg_3,fc1_topreal6,fc1_toprealb,fc1_xboole_0,fc1_yellow12,fc1_yellow13,fc23_toprealb,fc24_toprealb,fc2_borsuk_1,fc2_membered,fc2_pre_topc,fc2_rcomp_3,fc2_subset_1,fc2_topalg_3,fc2_topalg_5,fc2_yellow13,fc3_nat_1,fc3_pre_topc,fc3_rcomp_3,fc3_realset1,fc3_subset_1,fc3_topalg_5,fc3_xreal_0,fc3_yellow13,fc4_borsuk_2,fc4_nat_1,fc4_pre_topc,fc4_rcomp_3,fc4_subset_1,fc4_topalg_5,fc5_borsuk_1,fc5_borsuk_2,fc5_membered,fc5_pre_topc,fc5_rcomp_3,fc5_topalg_5,fc5_yellow12,fc6_borsuk_2,fc6_borsuk_5,fc6_int_1,fc6_membered,fc6_rcomp_3,fc6_yellow12,fc7_borsuk_2,fc7_borsuk_5,fc7_membered,fc7_rcomp_3,fc7_topreal6,fc7_xreal_0,fc8_membered,fc8_rcomp_3,fc8_topreal6,fc8_tops_1,fc8_xreal_0,fc9_membered,fc9_rcomp_3,fc9_tops_1,fc9_xreal_0,fraenkel_a_2_0_topalg_5,fraenkel_a_2_1_borsuk_1,fraenkel_a_3_0_borsuk_1,fraenkel_a_3_1_topalg_5,free_g1_pre_topc,l41_topalg_5,rc1_borsuk_5,rc1_funct_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_rcomp_3,rc1_realset1,rc1_subset_1,rc1_topreal6,rc1_toprealb,rc1_tops_1,rc1_tsep_1,rc1_xboole_0,rc1_xreal_0,rc2_borsuk_5,rc2_funct_1,rc2_goboard1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_rcomp_3,rc2_realset1,rc2_subset_1,rc2_tops_1,rc2_tsep_1,rc2_xboole_0,rc2_xreal_0,rc3_borsuk_2,rc3_borsuk_5,rc3_funct_1,rc3_nat_1,rc3_pre_topc,rc3_rcomp_3,rc3_struct_0,rc3_topalg_2,rc3_tops_1,rc3_xreal_0,rc4_pre_topc,rc4_rcomp_3,rc4_tops_1,rc4_xreal_0,rc5_pre_topc,rc5_rcomp_3,rc5_struct_0,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,redefinition_k12_mcart_1,redefinition_k13_binarith,redefinition_k1_nat_1,redefinition_k1_pcomps_1,redefinition_k1_struct_0,redefinition_k2_seq_1,redefinition_k3_finseq_1,redefinition_k3_pscomp_1,redefinition_k4_finseq_1,redefinition_k4_pre_topc,redefinition_k4_pscomp_1,redefinition_k4_relset_1,redefinition_k5_numbers,redefinition_k5_relset_1,redefinition_k5_setfam_1,redefinition_k5_topmetr,redefinition_k6_setfam_1,redefinition_k7_borsuk_1,redefinition_k8_borsuk_1,redefinition_k8_funct_2,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_funct_2__e56_40__topalg_5,s3_funct_2__e12_40__topalg_5,s3_funct_2__e9_40__topalg_5,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t106_zfmisc_1,t118_zfmisc_1,t119_zfmisc_1,t134_zfmisc_1,t156_relat_1,t17_tsep_1,t1_arithm,t1_jordan1,t1_numerals,t1_real,t1_subset,t1_xboole_1,t20_jgraph_2,t21_connsp_2,t26_connsp_3,t27_borsuk_3,t27_finseq_3,t27_topmetr,t27_tops_2,t2_real,t2_subset,t2_tarski,t35_connsp_3,t38_nat_1,t3_real,t3_subset,t3_topalg_5,t43_connsp_1,t45_borsuk_1,t46_borsuk_1,t4_real,t4_setfam_1,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t79_rcomp_3,t7_boole,t7_real,t80_rcomp_3,t8_boole,t8_real,t8_xreal_1,t92_zfmisc_1]), [file(topalg_5,t21_topalg_5)]). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc10_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_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_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_borsuk_2,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ( v1_xreal_0(A) & v1_xcmplx_0(A) ) ) ), file(borsuk_2,cc1_borsuk_2), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). 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_topalg_3,axiom,( ! [A] : ( l1_pre_topc(A) => ( v3_struct_0(A) => v1_connsp_1(A) ) ) ), file(topalg_3,cc1_topalg_3), []). 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_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_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc2_topalg_3,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v3_realset2(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & v1_connsp_1(A) ) ) ) ), file(topalg_3,cc2_topalg_3), []). 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_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_borsuk_5,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v1_rat_1(A) ) => ( ~ v1_xboole_0(A) & v1_xreal_0(A) & v1_xcmplx_0(A) ) ) ), file(borsuk_5,cc3_borsuk_5), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_topalg_2,axiom,( ! [A] : ( m1_pre_topc(A,k3_topmetr) => ( v2_pre_topc(A) & v1_borsuk_6(A) ) ) ), file(topalg_2,cc3_topalg_2), []). 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_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_topalg_2,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( v1_xboole_0(A) => v2_topalg_2(A) ) ) ), file(topalg_2,cc4_topalg_2), []). 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_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_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_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). 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_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_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_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_xboole_0(A) => v1_sppol_1(A) ) ) ), file(topreal6,cc8_topreal6), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), []). fof(cc8_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & v5_compts_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v4_compts_1(A) ) ) ) ), file(yellow13,cc8_yellow13), []). fof(cc9_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), []). fof(cc9_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_xboole_0(A) => v2_sppol_1(A) ) ) ), file(topreal6,cc9_topreal6), []). fof(cc9_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v1_urysohn1(A) & v4_compts_1(A) ) => ( v2_pre_topc(A) & v3_compts_1(A) ) ) ) ), file(yellow13,cc9_yellow13), []). fof(commutativity_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(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_compts_1,axiom,( ! [A] : ( l1_struct_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r1_compts_1(A,B,C) <=> r1_tarski(C,k5_setfam_1(u1_struct_0(A),B)) ) ) ) ) ), file(compts_1,d1_compts_1), []). fof(d1_finset_1,axiom,( ! [A] : ( v1_finset_1(A) <=> ? [B] : ( v1_relat_1(B) & v1_funct_1(B) & k2_relat_1(B) = A & r2_hidden(k1_relat_1(B),k5_ordinal2) ) ) ), file(finset_1,d1_finset_1), []). 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_rcomp_3,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_rcomp_3(B,A) <=> ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(C,B) => v2_connsp_1(C,A) ) ) ) ) ) ), file(rcomp_3,d1_rcomp_3), []). fof(d1_setfam_1,axiom,( ! [A,B] : ( ( A != k1_xboole_0 => ( B = k1_setfam_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ! [D] : ( r2_hidden(D,A) => r2_hidden(C,D) ) ) ) ) & ( A = k1_xboole_0 => ( B = k1_setfam_1(A) <=> B = k1_xboole_0 ) ) ) ), file(setfam_1,d1_setfam_1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_tops_2,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_tops_2(B,A) <=> ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(C,B) => v3_pre_topc(C,A) ) ) ) ) ) ), file(tops_2,d1_tops_2), []). fof(d2_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) => ( ( r1_pre_topc(k4_topmetr(A,B),C) & v1_tops_2(C,k4_topmetr(A,B)) & v1_rcomp_3(C,k4_topmetr(A,B)) & r1_xreal_0(A,B) ) => ! [D] : ( m2_finseq_1(D,k1_pcomps_1(k1_numbers)) => ( m1_rcomp_3(D,A,B,C) <=> ( r1_tarski(k5_relset_1(k5_numbers,k1_pcomps_1(k1_numbers),D),C) & k5_setfam_1(k1_numbers,k5_relset_1(k5_numbers,k1_pcomps_1(k1_numbers),D)) = k1_rcomp_1(A,B) & ! [E] : ( v4_ordinal2(E) => ( r1_xreal_0(1,E) => ( ~ ( r1_xreal_0(E,k3_finseq_1(D)) & v1_xboole_0(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)) ) & ( r1_xreal_0(k2_xcmplx_0(E,1),k3_finseq_1(D)) => ( r1_xreal_0(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,1)))) & r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,1)))) & ~ r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,1)))) ) ) & ( r1_xreal_0(k2_xcmplx_0(E,2),k3_finseq_1(D)) => r1_xreal_0(k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,E)),k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(E,2)))) ) ) ) ) & ( r2_hidden(k1_rcomp_1(A,B),C) => D = k13_binarith(k1_zfmisc_1(k1_numbers),k1_rcomp_1(A,B)) ) & ( ~ r2_hidden(k1_rcomp_1(A,B),C) => ( ? [E] : ( v1_xreal_0(E) & ~ r1_xreal_0(E,A) & r1_xreal_0(E,B) & k1_funct_1(D,1) = k1_rcomp_2(A,E) ) & ? [E] : ( v1_xreal_0(E) & r1_xreal_0(A,E) & ~ r1_xreal_0(B,E) & k1_funct_1(D,k3_finseq_1(D)) = k2_rcomp_2(E,B) ) & ! [E] : ( v4_ordinal2(E) => ~ ( ~ r1_xreal_0(E,1) & ~ r1_xreal_0(k3_finseq_1(D),E) & ! [F] : ( v1_xreal_0(F) => ! [G] : ( v1_xreal_0(G) => ~ ( r1_xreal_0(A,F) & ~ r1_xreal_0(G,F) & r1_xreal_0(G,B) & k1_funct_1(D,E) = k2_rcomp_1(F,G) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(rcomp_3,d2_rcomp_3), []). fof(d2_zfmisc_1,axiom,( ! [A,B,C] : ( C = k2_zfmisc_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E,F] : ( r2_hidden(E,A) & r2_hidden(F,B) & D = k4_tarski(E,F) ) ) ) ), file(zfmisc_1,d2_zfmisc_1), []). fof(d3_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,d3_pre_topc), []). fof(d3_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) => ! [D] : ( m1_rcomp_3(D,A,B,C) => ( ( r1_pre_topc(k4_topmetr(A,B),C) & v1_tops_2(C,k4_topmetr(A,B)) & v1_rcomp_3(C,k4_topmetr(A,B)) & r1_xreal_0(A,B) ) => ! [E] : ( m2_finseq_1(E,k1_numbers) => ( m2_rcomp_3(E,A,B,C,D) <=> ( k3_finseq_1(E) = k1_nat_1(k3_finseq_1(D),1) & k2_seq_1(k5_numbers,k1_numbers,E,1) = A & k2_seq_1(k5_numbers,k1_numbers,E,k3_finseq_1(E)) = B & ! [F] : ( v4_ordinal2(F) => ( r1_xreal_0(1,F) => ( r1_xreal_0(k3_finseq_1(E),k2_xcmplx_0(F,1)) | r2_hidden(k2_seq_1(k5_numbers,k1_numbers,E,k2_xcmplx_0(F,1)),k2_rcomp_1(k4_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,k2_xcmplx_0(F,1))),k3_pscomp_1(k4_finseq_4(k5_numbers,k1_pcomps_1(k1_numbers),D,F)))) ) ) ) ) ) ) ) ) ) ) ) ), file(rcomp_3,d3_rcomp_3), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). 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_borsuk_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_pre_topc(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ( C = k6_borsuk_1(A,B) <=> ( u1_struct_0(C) = k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)) & u1_pre_topc(C) = a_3_0_borsuk_1(A,B,C) ) ) ) ) ) ), file(borsuk_1,d5_borsuk_1), []). fof(d5_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d7_compts_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v6_compts_1(B,A) <=> ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ~ ( r1_compts_1(A,C,B) & v1_tops_2(C,A) & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ~ ( r1_tarski(D,C) & r1_compts_1(A,D,B) & v1_finset_1(D) ) ) ) ) ) ) ) ), file(compts_1,d7_compts_1), []). fof(d7_connsp_3,axiom,( ! [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,B) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( D = k6_connsp_3(A,B,C) <=> ! [E] : ( m1_subset_1(E,u1_struct_0(k3_pre_topc(A,B))) => ( E = C => D = k1_connsp_1(k3_pre_topc(A,B),E) ) ) ) ) ) ) ) ) ), file(connsp_3,d7_connsp_3), []). fof(d8_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( r1_pre_topc(A,B) <=> k2_pre_topc(A) = k5_setfam_1(u1_struct_0(A),B) ) ) ) ), file(pre_topc,d8_pre_topc), []). 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_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => m1_subset_1(k12_mcart_1(A,B,C,D),k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(mcart_1,k12_mcart_1), []). fof(dt_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(1,A)) ) ), file(binarith,k13_binarith), []). 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_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). 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_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_pcomps_1,axiom,( ! [A] : m1_subset_1(k1_pcomps_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ), file(pcomps_1,k1_pcomps_1), []). fof(dt_k1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k1_rcomp_1), []). fof(dt_k1_rcomp_2,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_2,k1_rcomp_2), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_setfam_1,axiom,( $true ), file(setfam_1,k1_setfam_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_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_k22_borsuk_1,axiom,( l1_pre_topc(k22_borsuk_1) ), file(borsuk_1,k22_borsuk_1), []). fof(dt_k23_borsuk_1,axiom,( m1_subset_1(k23_borsuk_1,u1_struct_0(k22_borsuk_1)) ), file(borsuk_1,k23_borsuk_1), []). fof(dt_k2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k2_pre_topc), []). fof(dt_k2_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k2_rcomp_1), []). fof(dt_k2_rcomp_2,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k2_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_2,k2_rcomp_2), []). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), []). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), []). fof(dt_k2_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k2_seq_4(A)) ) ), file(seq_4,k2_seq_4), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), []). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), []). fof(dt_k3_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k3_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k3_pscomp_1), []). fof(dt_k3_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k3_seq_4(A)) ) ), file(seq_4,k3_seq_4), []). fof(dt_k3_tarski,axiom,( $true ), file(tarski,k3_tarski), []). fof(dt_k3_topmetr,axiom, ( v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & l1_pre_topc(k3_topmetr) ), file(topmetr,k3_topmetr), []). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), []). 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_finseq_4,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k4_finseq_4(A,B,C,D),B) ) ), file(finseq_4,k4_finseq_4), []). fof(dt_k4_pre_topc,axiom,( ! [A,B,C,D] : ( ( l1_struct_0(A) & l1_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => m1_subset_1(k4_pre_topc(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) ), file(pre_topc,k4_pre_topc), []). fof(dt_k4_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => m1_subset_1(k4_pscomp_1(A),k1_numbers) ) ), file(pscomp_1,k4_pscomp_1), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_topmetr,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( ~ v3_struct_0(k4_topmetr(A,B)) & v1_pre_topc(k4_topmetr(A,B)) & m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ), file(topmetr,k4_topmetr), []). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), []). 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_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) ), file(relset_1,k5_relset_1), []). fof(dt_k5_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) ), file(setfam_1,k5_setfam_1), []). fof(dt_k5_topmetr,axiom, ( v1_pre_topc(k5_topmetr) & m1_pre_topc(k5_topmetr,k3_topmetr) ), file(topmetr,k5_topmetr), []). fof(dt_k6_borsuk_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v2_pre_topc(B) & l1_pre_topc(B) ) => ( v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & l1_pre_topc(k6_borsuk_1(A,B)) ) ) ), file(borsuk_1,k6_borsuk_1), []). fof(dt_k6_connsp_3,axiom,( ! [A,B,C] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k6_connsp_3(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(connsp_3,k6_connsp_3), []). fof(dt_k6_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => m1_subset_1(k6_setfam_1(A,B),k1_zfmisc_1(A)) ) ), file(setfam_1,k6_setfam_1), []). fof(dt_k7_borsuk_1,axiom,( ! [A,B,C,D] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) ) => m1_subset_1(k7_borsuk_1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) ), file(borsuk_1,k7_borsuk_1), []). fof(dt_k8_borsuk_1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,u1_struct_0(A)) & m1_subset_1(D,u1_struct_0(B)) ) => m1_subset_1(k8_borsuk_1(A,B,C,D),u1_struct_0(k6_borsuk_1(A,B))) ) ), file(borsuk_1,k8_borsuk_1), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_toprealb,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_pre_topc(k8_toprealb(A),k15_euclid(A)) ) ), file(toprealb,k8_toprealb), []). fof(dt_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_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_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), []). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), []). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), []). fof(dt_m1_rcomp_3,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) ) => ! [D] : ( m1_rcomp_3(D,A,B,C) => m2_finseq_1(D,k1_pcomps_1(k1_numbers)) ) ) ), file(rcomp_3,m1_rcomp_3), []). 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_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), []). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), []). fof(dt_m2_rcomp_3,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) & m1_rcomp_3(D,A,B,C) ) => ! [E] : ( m2_rcomp_3(E,A,B,C,D) => m2_finseq_1(E,k1_numbers) ) ) ), file(rcomp_3,m2_rcomp_3), []). 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_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_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_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), []). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), []). fof(existence_m1_rcomp_3,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) ) => ? [D] : m1_rcomp_3(D,A,B,C) ) ), file(rcomp_3,m1_rcomp_3), []). 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_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), []). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), []). fof(existence_m2_rcomp_3,axiom,( ! [A,B,C,D] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) & m1_rcomp_3(D,A,B,C) ) => ? [E] : m2_rcomp_3(E,A,B,C,D) ) ), file(rcomp_3,m2_rcomp_3), []). 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_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_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_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_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(fc14_membered,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), []). fof(fc15_membered,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), []). fof(fc16_membered,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), []). fof(fc17_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v3_struct_0(k4_topmetr(A,A)) & v1_pre_topc(k4_topmetr(A,A)) & v2_pre_topc(k4_topmetr(A,A)) & v2_compts_1(k4_topmetr(A,A)) & v6_group_1(k4_topmetr(A,A)) & v3_realset2(k4_topmetr(A,A)) & v10_pscomp_1(k4_topmetr(A,A)) & v1_borsuk_6(k4_topmetr(A,A)) ) ) ), file(rcomp_3,fc17_rcomp_3), []). fof(fc18_borsuk_5,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v1_connsp_1(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) & v1_borsuk_2(k22_borsuk_1) & v1_urysohn1(k22_borsuk_1) ), file(borsuk_5,fc18_borsuk_5), []). fof(fc1_borsuk_5,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & v1_connsp_1(k3_topmetr) & v2_t_0topsp(k3_topmetr) & v3_compts_1(k3_topmetr) & v1_borsuk_2(k3_topmetr) & v1_urysohn1(k3_topmetr) ), file(borsuk_5,fc1_borsuk_5), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_topalg_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) ) => ( v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v2_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v1_connsp_1(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ), file(topalg_3,fc1_topalg_3), []). 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_toprealb,axiom, ( ~ v1_xboole_0(k2_rcomp_1(0,1)) & v1_membered(k2_rcomp_1(0,1)) & v2_membered(k2_rcomp_1(0,1)) & v3_rcomp_1(k2_rcomp_1(0,1)) ), file(toprealb,fc1_toprealb), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). fof(fc1_yellow12,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_xboole_0(k3_tarski(A)) & v1_membered(k3_tarski(A)) & v2_membered(k3_tarski(A)) & v3_membered(k3_tarski(A)) & v4_membered(k3_tarski(A)) & v5_membered(k3_tarski(A)) ) ) ), file(yellow12,fc1_yellow12), []). 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(fc23_toprealb,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ( ~ v3_struct_0(k8_toprealb(A)) & v2_pre_topc(k8_toprealb(A)) & v1_monoid_0(k8_toprealb(A)) & v2_monoid_0(k8_toprealb(A)) ) ) ), file(toprealb,fc23_toprealb), []). fof(fc24_toprealb,axiom, ( ~ v3_struct_0(k8_toprealb(2)) & v2_pre_topc(k8_toprealb(2)) & v2_compts_1(k8_toprealb(2)) & v10_pscomp_1(k8_toprealb(2)) & v1_borsuk_2(k8_toprealb(2)) & v1_monoid_0(k8_toprealb(2)) & v2_monoid_0(k8_toprealb(2)) & v1_toprealb(k8_toprealb(2)) ), file(toprealb,fc24_toprealb), []). fof(fc2_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) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) ) ) ), file(borsuk_1,fc2_borsuk_1), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(k2_pre_topc(A)) ) ), file(pre_topc,fc2_pre_topc), []). fof(fc2_rcomp_3,axiom,( ! [A] : ( ( v1_borsuk_6(A) & l1_struct_0(A) ) => ( v1_membered(u1_struct_0(A)) & v2_membered(u1_struct_0(A)) ) ) ), file(rcomp_3,fc2_rcomp_3), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_topalg_3,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) & v2_pre_topc(B) & v1_connsp_1(B) & l1_pre_topc(B) ) => ( v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v1_connsp_1(k6_borsuk_1(A,B)) ) ) ), file(topalg_3,fc2_topalg_3), []). 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_yellow13,axiom,( ! [A] : ( ( v3_realset2(A) & l1_struct_0(A) ) => v1_realset1(u1_struct_0(A)) ) ), file(yellow13,fc2_yellow13), []). fof(fc3_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), []). fof(fc3_pre_topc,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k3_pre_topc(A,B)) & v1_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc3_pre_topc), []). fof(fc3_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_2(A,B)) & v2_membered(k1_rcomp_2(A,B)) & v1_seq_4(k1_rcomp_2(A,B)) & v2_seq_4(k1_rcomp_2(A,B)) & v3_seq_4(k1_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc3_rcomp_3), []). 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_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_topalg_5,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v3_pre_topc(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) & v1_tsep_1(k3_pre_topc(A,B),A) ) ) ), file(topalg_5,fc3_topalg_5), []). fof(fc3_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), []). fof(fc3_yellow13,axiom,( ! [A] : ( ( v2_compts_1(A) & l1_pre_topc(A) ) => ( v1_tops_1(k2_pre_topc(A),A) & v6_compts_1(k2_pre_topc(A),A) ) ) ), file(yellow13,fc3_yellow13), []). fof(fc4_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_t_0topsp(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & v2_t_0topsp(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v2_t_0topsp(k6_borsuk_1(A,B)) ) ) ), file(borsuk_2,fc4_borsuk_2), []). fof(fc4_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), []). fof(fc4_pre_topc,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc4_pre_topc), []). fof(fc4_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_2(A,B)) & v2_membered(k2_rcomp_2(A,B)) & v1_seq_4(k2_rcomp_2(A,B)) & v2_seq_4(k2_rcomp_2(A,B)) & v3_seq_4(k2_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc4_rcomp_3), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc4_topalg_5,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v1_connsp_2(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) ), file(topalg_5,fc4_topalg_5), []). fof(fc5_borsuk_1,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) ), file(borsuk_1,fc5_borsuk_1), []). fof(fc5_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & v1_urysohn1(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v1_urysohn1(k6_borsuk_1(A,B)) ) ) ), file(borsuk_2,fc5_borsuk_2), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v4_pre_topc(k2_pre_topc(A),A) ) ), file(pre_topc,fc5_pre_topc), []). fof(fc5_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_1(A,B)) & v2_membered(k2_rcomp_1(A,B)) & v1_seq_4(k2_rcomp_1(A,B)) & v2_seq_4(k2_rcomp_1(A,B)) & v3_seq_4(k2_rcomp_1(A,B)) & v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ), file(rcomp_3,fc5_rcomp_3), []). fof(fc5_topalg_5,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v3_pre_topc(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ( ~ v3_struct_0(k3_pre_topc(k5_topmetr,A)) & v1_pre_topc(k3_pre_topc(k5_topmetr,A)) & v2_pre_topc(k3_pre_topc(k5_topmetr,A)) & v1_connsp_2(k3_pre_topc(k5_topmetr,A)) & v1_tsep_1(k3_pre_topc(k5_topmetr,A),k5_topmetr) ) ) ), file(topalg_5,fc5_topalg_5), []). fof(fc5_yellow12,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ( ~ v3_struct_0(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ), file(yellow12,fc5_yellow12), []). fof(fc6_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v3_compts_1(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & v3_compts_1(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v2_t_0topsp(k6_borsuk_1(A,B)) & v3_compts_1(k6_borsuk_1(A,B)) & v3_yellow_8(k6_borsuk_1(A,B)) ) ) ), file(borsuk_2,fc6_borsuk_2), []). fof(fc6_borsuk_5,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & ~ v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(borsuk_5,fc6_borsuk_5), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc6_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v1_seq_4(k1_rcomp_1(A,B)) & v2_seq_4(k1_rcomp_1(A,B)) & v3_seq_4(k1_rcomp_1(A,B)) & v1_rcomp_1(k1_rcomp_1(A,B)) & v2_rcomp_1(k1_rcomp_1(A,B)) & v1_jct_misc(k1_rcomp_1(A,B)) ) ) ), file(rcomp_3,fc6_rcomp_3), []). fof(fc6_yellow12,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v2_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ), file(yellow12,fc6_yellow12), []). fof(fc7_borsuk_2,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) ), file(borsuk_2,fc7_borsuk_2), []). fof(fc7_borsuk_5,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & ~ v1_rat_1(k2_xcmplx_0(B,A)) ) ) ), file(borsuk_5,fc7_borsuk_5), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). fof(fc7_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_2(A,B)) & v2_membered(k1_rcomp_2(A,B)) & v1_seq_4(k1_rcomp_2(A,B)) & v2_seq_4(k1_rcomp_2(A,B)) & v3_seq_4(k1_rcomp_2(A,B)) & v1_jct_misc(k1_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc7_rcomp_3), []). fof(fc7_topreal6,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & v3_compts_1(k3_topmetr) ), file(topreal6,fc7_topreal6), []). fof(fc7_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), []). fof(fc8_membered,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), []). fof(fc8_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_2(A,B)) & v2_membered(k2_rcomp_2(A,B)) & v1_seq_4(k2_rcomp_2(A,B)) & v2_seq_4(k2_rcomp_2(A,B)) & v3_seq_4(k2_rcomp_2(A,B)) & v1_jct_misc(k2_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc8_rcomp_3), []). fof(fc8_topreal6,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v1_seq_4(k1_rcomp_1(A,B)) & v2_seq_4(k1_rcomp_1(A,B)) & v3_seq_4(k1_rcomp_1(A,B)) & v1_rcomp_1(k1_rcomp_1(A,B)) & v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ), file(topreal6,fc8_topreal6), []). fof(fc8_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v3_pre_topc(k2_pre_topc(A),A) & v4_pre_topc(k2_pre_topc(A),A) ) ) ), file(tops_1,fc8_tops_1), []). fof(fc8_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), []). fof(fc9_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_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_1(A,B)) & v2_membered(k2_rcomp_1(A,B)) & v1_seq_4(k2_rcomp_1(A,B)) & v2_seq_4(k2_rcomp_1(A,B)) & v3_seq_4(k2_rcomp_1(A,B)) & v3_rcomp_1(k2_rcomp_1(A,B)) & v1_jct_misc(k2_rcomp_1(A,B)) ) ) ), file(rcomp_3,fc9_rcomp_3), []). fof(fc9_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => v1_tops_1(k2_pre_topc(A),A) ) ), file(tops_1,fc9_tops_1), []). fof(fc9_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(fraenkel_a_2_0_topalg_5,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,k5_topmetr))))) ) => ( r2_hidden(A,a_2_0_topalg_5(B,C)) <=> ? [D] : ( v3_pre_topc(D,k5_topmetr) & v2_connsp_1(D,k5_topmetr) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k5_topmetr))) & A = D & ? [E] : ( v3_pre_topc(E,B) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) & r2_hidden(k7_borsuk_1(B,k5_topmetr,E,D),C) ) ) ) ) ), file(topalg_5,a_2_0_topalg_5), []). fof(fraenkel_a_2_1_borsuk_1,axiom,( ! [A,B,C] : ( ( v2_pre_topc(B) & l1_pre_topc(B) & v2_pre_topc(C) & l1_pre_topc(C) ) => ( r2_hidden(A,a_2_1_borsuk_1(B,C)) <=> ? [D,E] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(C))) & A = k12_mcart_1(u1_struct_0(B),u1_struct_0(C),D,E) & r2_hidden(D,u1_pre_topc(B)) & r2_hidden(E,u1_pre_topc(C)) ) ) ) ), file(borsuk_1,a_2_1_borsuk_1), []). fof(fraenkel_a_3_0_borsuk_1,axiom,( ! [A,B,C,D] : ( ( v2_pre_topc(B) & l1_pre_topc(B) & v2_pre_topc(C) & l1_pre_topc(C) & v1_pre_topc(D) & v2_pre_topc(D) & l1_pre_topc(D) ) => ( r2_hidden(A,a_3_0_borsuk_1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(D)))) & A = k5_setfam_1(u1_struct_0(D),E) & r1_tarski(E,a_2_1_borsuk_1(B,C)) ) ) ) ), file(borsuk_1,a_3_0_borsuk_1), []). fof(fraenkel_a_3_1_topalg_5,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,u1_struct_0(B)) & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(B,k5_topmetr))))) ) => ( r2_hidden(A,a_3_1_topalg_5(B,C,D)) <=> ? [E] : ( v3_pre_topc(E,B) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) & A = E & r2_hidden(C,E) & ? [F] : ( v3_pre_topc(F,k5_topmetr) & m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_topmetr))) & r2_hidden(k7_borsuk_1(B,k5_topmetr,E,F),D) ) ) ) ) ), file(topalg_5,a_3_1_topalg_5), []). 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(l41_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) => ! [D] : ( m1_rcomp_3(D,A,B,C) => ! [E] : ( m2_rcomp_3(E,A,B,C,D) => ~ ( r1_pre_topc(k4_topmetr(A,B),C) & v1_tops_2(C,k4_topmetr(A,B)) & v1_rcomp_3(C,k4_topmetr(A,B)) & r1_xreal_0(A,B) & v1_xboole_0(E) ) ) ) ) ) ) ), file(topalg_5,l41_topalg_5), []). fof(rc1_borsuk_5,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v1_connsp_1(A) ) ), file(borsuk_5,rc1_borsuk_5), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_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_rcomp_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_membered(A) & v2_membered(A) & v1_seq_4(A) & v2_seq_4(A) ) ), file(rcomp_3,rc1_rcomp_3), []). fof(rc1_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(realset1,rc1_realset1), []). 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_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_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_tsep_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_tsep_1(B,A) ) ) ), file(tsep_1,rc1_tsep_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_borsuk_5,axiom,( ? [A] : ( v1_xreal_0(A) & v1_xcmplx_0(A) & ~ v1_rat_1(A) ) ), file(borsuk_5,rc2_borsuk_5), []). 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_goboard1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_finseq_1(B,A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ) ), file(goboard1,rc2_goboard1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_pcomps_1,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v3_compts_1(A) ) ), file(pcomps_1,rc2_pcomps_1), []). fof(rc2_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), []). fof(rc2_rcomp_3,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v1_seq_4(A) & v2_seq_4(A) & v3_seq_4(A) & v3_rcomp_1(A) & v1_jct_misc(A) ) ), file(rcomp_3,rc2_rcomp_3), []). 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_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_tsep_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_tsep_1(B,A) ) ) ), file(tsep_1,rc2_tsep_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc2_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), []). fof(rc3_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_5,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xreal_0(A) & v1_xcmplx_0(A) & v1_rat_1(A) ) ), file(borsuk_5,rc3_borsuk_5), []). 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_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_rcomp_3,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & ~ v3_seq_4(A) & v2_rcomp_1(A) & v3_rcomp_1(A) & v1_jct_misc(A) ) ), file(rcomp_3,rc3_rcomp_3), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_topalg_2,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) & ~ v1_xboole_0(A) & v2_topalg_2(A) ) ), file(topalg_2,rc3_topalg_2), []). 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_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_rcomp_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v2_connsp_1(B,A) ) ) ), file(rcomp_3,rc4_rcomp_3), []). fof(rc4_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) ) ) ), file(tops_1,rc4_tops_1), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), []). fof(rc5_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(rc7_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc7_pre_topc), []). fof(redefinition_k12_mcart_1,axiom,( ! [A,B,C,D] : ( ( m1_subset_1(C,k1_zfmisc_1(A)) & m1_subset_1(D,k1_zfmisc_1(B)) ) => k12_mcart_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ), file(mcart_1,k12_mcart_1), []). fof(redefinition_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k13_binarith(A,B) = k5_finseq_1(B) ) ), file(binarith,k13_binarith), []). fof(redefinition_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), []). fof(redefinition_k1_pcomps_1,axiom,( ! [A] : k1_pcomps_1(A) = k1_zfmisc_1(A) ), file(pcomps_1,k1_pcomps_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_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), []). fof(redefinition_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), []). fof(redefinition_k3_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k3_pscomp_1(A) = k2_seq_4(A) ) ), file(pscomp_1,k3_pscomp_1), []). fof(redefinition_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), []). fof(redefinition_k4_pre_topc,axiom,( ! [A,B,C,D] : ( ( l1_struct_0(A) & l1_struct_0(B) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => k4_pre_topc(A,B,C,D) = k9_relat_1(C,D) ) ), file(pre_topc,k4_pre_topc), []). fof(redefinition_k4_pscomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => k4_pscomp_1(A) = k3_seq_4(A) ) ), file(pscomp_1,k4_pscomp_1), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k5_relset_1(A,B,C) = k2_relat_1(C) ) ), file(relset_1,k5_relset_1), []). fof(redefinition_k5_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => k5_setfam_1(A,B) = k3_tarski(B) ) ), file(setfam_1,k5_setfam_1), []). fof(redefinition_k5_topmetr,axiom,( k5_topmetr = k22_borsuk_1 ), file(topmetr,k5_topmetr), []). fof(redefinition_k6_setfam_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => k6_setfam_1(A,B) = k1_setfam_1(B) ) ), file(setfam_1,k6_setfam_1), []). fof(redefinition_k7_borsuk_1,axiom,( ! [A,B,C,D] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) ) => k7_borsuk_1(A,B,C,D) = k2_zfmisc_1(C,D) ) ), file(borsuk_1,k7_borsuk_1), []). fof(redefinition_k8_borsuk_1,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & m1_subset_1(C,u1_struct_0(A)) & m1_subset_1(D,u1_struct_0(B)) ) => k8_borsuk_1(A,B,C,D) = k4_tarski(C,D) ) ), file(borsuk_1,k8_borsuk_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), []). fof(redefinition_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), []). fof(s1_funct_2__e56_40__topalg_5,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) & v1_relat_1(D) & v1_funct_1(D) ) => ( ! [E] : ~ ( r2_hidden(E,k1_relat_1(D)) & ! [F] : ~ ( r2_hidden(F,C) & ? [G] : ( v3_pre_topc(G,A) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(A))) & ? [H] : ( ~ v1_xboole_0(H) & v3_pre_topc(H,k5_topmetr) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k5_topmetr))) & r2_hidden(B,G) & k1_funct_1(D,E) = k7_borsuk_1(A,k5_topmetr,G,H) & F = G ) ) ) ) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k1_relat_1(D),C) & m2_relset_1(E,k1_relat_1(D),C) & ! [F] : ( r2_hidden(F,k1_relat_1(D)) => ? [I] : ( v3_pre_topc(I,A) & m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A))) & ? [J] : ( ~ v1_xboole_0(J) & v3_pre_topc(J,k5_topmetr) & m1_subset_1(J,k1_zfmisc_1(u1_struct_0(k5_topmetr))) & r2_hidden(B,I) & k1_funct_1(D,F) = k7_borsuk_1(A,k5_topmetr,I,J) & k1_funct_1(E,F) = I ) ) ) ) ) ) ), file(topalg_5,s1_funct_2__e56_40__topalg_5), []). fof(s3_funct_2__e12_40__topalg_5,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(2))) & v5_pre_topc(B,k6_borsuk_1(A,k5_topmetr),k8_toprealb(2)) & m2_relset_1(B,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(2))) & m1_subset_1(C,u1_struct_0(A)) & ~ v1_xboole_0(D) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),D) & m2_relset_1(E,u1_struct_0(k5_topmetr),D) ) => ( ! [F] : ( m1_subset_1(F,u1_struct_0(k5_topmetr)) => ? [G] : ( m1_subset_1(G,k1_pcomps_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(k5_topmetr)))) & ? [H] : ( v3_pre_topc(H,A) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A))) & ? [I] : ( v3_pre_topc(I,k5_topmetr) & v2_connsp_1(I,k5_topmetr) & m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k5_topmetr))) & r2_hidden(C,H) & r2_hidden(F,I) & r1_tarski(k4_pre_topc(k6_borsuk_1(A,k5_topmetr),k8_toprealb(2),B,k7_borsuk_1(A,k5_topmetr,H,I)),k1_funct_1(E,F)) & G = k7_borsuk_1(A,k5_topmetr,H,I) ) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),k1_pcomps_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(k5_topmetr)))) & m2_relset_1(F,u1_struct_0(k5_topmetr),k1_pcomps_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(k5_topmetr)))) & ! [G] : ( m1_subset_1(G,u1_struct_0(k5_topmetr)) => ? [J] : ( v3_pre_topc(J,A) & m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A))) & ? [K] : ( v3_pre_topc(K,k5_topmetr) & v2_connsp_1(K,k5_topmetr) & m1_subset_1(K,k1_zfmisc_1(u1_struct_0(k5_topmetr))) & r2_hidden(C,J) & r2_hidden(G,K) & r1_tarski(k4_pre_topc(k6_borsuk_1(A,k5_topmetr),k8_toprealb(2),B,k7_borsuk_1(A,k5_topmetr,J,K)),k1_funct_1(E,G)) & k8_funct_2(u1_struct_0(k5_topmetr),k1_pcomps_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(k5_topmetr))),F,G) = k7_borsuk_1(A,k5_topmetr,J,K) ) ) ) ) ) ) ), file(topalg_5,s3_funct_2__e12_40__topalg_5), []). fof(s3_funct_2__e9_40__topalg_5,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(2))) & v5_pre_topc(B,k6_borsuk_1(A,k5_topmetr),k8_toprealb(2)) & m2_relset_1(B,u1_struct_0(k6_borsuk_1(A,k5_topmetr)),u1_struct_0(k8_toprealb(2))) & m1_subset_1(C,u1_struct_0(A)) & ~ v1_xboole_0(D) ) => ( ! [E] : ( m1_subset_1(E,u1_struct_0(k5_topmetr)) => ? [F] : ( m1_subset_1(F,D) & ? [G] : ( v3_pre_topc(G,k8_toprealb(2)) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k8_toprealb(2)))) & r2_hidden(G,D) & r2_hidden(k1_funct_1(B,k4_tarski(C,E)),G) & F = G ) ) ) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),D) & m2_relset_1(E,u1_struct_0(k5_topmetr),D) & ! [F] : ( m1_subset_1(F,u1_struct_0(k5_topmetr)) => ? [H] : ( v3_pre_topc(H,k8_toprealb(2)) & m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k8_toprealb(2)))) & r2_hidden(H,D) & r2_hidden(k1_funct_1(B,k4_tarski(C,F)),H) & k8_funct_2(u1_struct_0(k5_topmetr),D,E,F) = H ) ) ) ) ) ), file(topalg_5,s3_funct_2__e9_40__topalg_5), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(t106_zfmisc_1,axiom,( ! [A,B,C,D] : ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D)) <=> ( r2_hidden(A,C) & r2_hidden(B,D) ) ) ), file(zfmisc_1,t106_zfmisc_1), []). fof(t118_zfmisc_1,axiom,( ! [A,B,C] : ( r1_tarski(A,B) => ( r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C)) & r1_tarski(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ) ), file(zfmisc_1,t118_zfmisc_1), []). fof(t119_zfmisc_1,axiom,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) ) => r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ), file(zfmisc_1,t119_zfmisc_1), []). fof(t134_zfmisc_1,axiom,( ! [A,B,C,D] : ( k2_zfmisc_1(A,B) = k2_zfmisc_1(C,D) => ( A = k1_xboole_0 | B = k1_xboole_0 | ( A = C & B = D ) ) ) ), file(zfmisc_1,t134_zfmisc_1), []). fof(t156_relat_1,axiom,( ! [A,B,C] : ( v1_relat_1(C) => ( r1_tarski(A,B) => r1_tarski(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ), file(relat_1,t156_relat_1), []). 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(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). 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(t20_jgraph_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v5_pre_topc(C,A,B) <=> ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) => ~ ( r2_hidden(k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D),E) & v3_pre_topc(E,B) & ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r2_hidden(D,F) & v3_pre_topc(F,A) & r1_tarski(k4_pre_topc(A,B,C,F),E) ) ) ) ) ) ) ) ) ) ), file(jgraph_2,t20_jgraph_2), []). 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(t26_connsp_3,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,B) => r2_hidden(C,k6_connsp_3(A,B,C)) ) ) ) ) ), file(connsp_3,t26_connsp_3), []). fof(t27_borsuk_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( ( v6_compts_1(C,A) & v6_compts_1(D,B) ) => ( v6_compts_1(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B)) & m1_subset_1(k7_borsuk_1(A,B,C,D),k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) ) ) ) ) ) ) ), file(borsuk_3,t27_borsuk_3), []). fof(t27_finseq_3,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(B,k4_finseq_1(A)) <=> ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ), file(finseq_3,t27_finseq_3), []). fof(t27_topmetr,axiom,( k4_topmetr(0,1) = k22_borsuk_1 ), file(topmetr,t27_topmetr), []). fof(t27_tops_2,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( ( v1_tops_2(B,A) & v1_finset_1(B) ) => v3_pre_topc(k6_setfam_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_2,t27_tops_2), []). 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(t35_connsp_3,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,B) => v2_connsp_1(k6_connsp_3(A,B,C),A) ) ) ) ) ), file(connsp_3,t35_connsp_3), []). fof(t38_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), []). 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_topalg_5,axiom,( ! [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,B) => r1_tarski(k6_connsp_3(A,B,C),B) ) ) ) ) ), file(topalg_5,t3_topalg_5), []). 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(t45_borsuk_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) => ( v3_pre_topc(C,k6_borsuk_1(A,B)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))) & C = k5_setfam_1(u1_struct_0(k6_borsuk_1(A,B)),D) & ! [E] : ~ ( r2_hidden(E,D) & ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B))) => ~ ( E = k12_mcart_1(u1_struct_0(A),u1_struct_0(B),F,G) & v3_pre_topc(F,A) & v3_pre_topc(G,B) ) ) ) ) ) ) ) ) ) ), file(borsuk_1,t45_borsuk_1), []). fof(t46_borsuk_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( ( v3_pre_topc(C,A) & v3_pre_topc(D,B) ) => v3_pre_topc(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B)) ) ) ) ) ) ), file(borsuk_1,t46_borsuk_1), []). 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_setfam_1,axiom,( ! [A,B] : ( r2_hidden(A,B) => r1_tarski(k1_setfam_1(B),A) ) ), file(setfam_1,t4_setfam_1), []). 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(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_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(t79_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) => ! [D] : ( m1_rcomp_3(D,A,B,C) => ! [E] : ( m2_rcomp_3(E,A,B,C,D) => ( ( r1_pre_topc(k4_topmetr(A,B),C) & v1_tops_2(C,k4_topmetr(A,B)) & v1_rcomp_3(C,k4_topmetr(A,B)) ) => ( r1_xreal_0(B,A) | v1_seqm_3(E) ) ) ) ) ) ) ) ), file(rcomp_3,t79_rcomp_3), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t80_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v4_ordinal2(C) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B))))) => ! [E] : ( m1_rcomp_3(E,A,B,D) => ! [F] : ( m2_rcomp_3(F,A,B,D,E) => ( ( r1_pre_topc(k4_topmetr(A,B),D) & v1_tops_2(D,k4_topmetr(A,B)) & v1_rcomp_3(D,k4_topmetr(A,B)) & r1_xreal_0(A,B) & r1_xreal_0(1,C) ) => ( r1_xreal_0(k3_finseq_1(F),C) | r1_tarski(k1_rcomp_1(k2_seq_1(k5_numbers,k1_numbers,F,C),k2_seq_1(k5_numbers,k1_numbers,F,k2_xcmplx_0(C,1))),k1_funct_1(E,C)) ) ) ) ) ) ) ) ) ), file(rcomp_3,t80_rcomp_3), []). 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_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), []). fof(t92_zfmisc_1,axiom,( ! [A,B] : ( r2_hidden(A,B) => r1_tarski(A,k3_tarski(B)) ) ), file(zfmisc_1,t92_zfmisc_1), []).