% Mizar problem: t19_nagata_2,nagata_2,1480,23 fof(t19_nagata_2,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( ( v4_compts_1(A) & v1_urysohn1(A) & ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & v6_nagata_1(B,A) ) ) <=> v3_pcomps_1(A) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc11_topreal6,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_fraenkel,cc1_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc1_topreal6,cc1_tops_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc2_topreal6,cc2_tops_1,cc2_xreal_0,cc3_membered,cc3_nat_1,cc3_tops_1,cc3_xreal_0,cc4_membered,cc4_tops_1,cc4_xreal_0,cc5_tops_1,cc5_xreal_0,cc6_membered,cc6_tops_1,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k11_seq_1,commutativity_k1_nat_1,commutativity_k1_square_1,commutativity_k2_tarski,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_square_1,commutativity_k3_xboole_0,commutativity_k5_seq_1,connectedness_r1_xreal_0,d11_finseq_1,d12_funct_1,d1_absvalue,d1_binop_1,d1_finsop_1,d1_funct_2,d1_square_1,d1_xboole_0,d2_cantor_1,d2_nagata_2,d2_pcomps_1,d2_setwiseo,d2_xboole_0,d2_zfmisc_1,d3_finseq_1,d3_funct_2,d3_nagata_1,d3_nagata_2,d3_pcomps_1,d3_tarski,d4_finseq_1,d4_funct_2,d4_tarski,d4_xboole_0,d5_borsuk_1,d5_funct_1,d5_pre_topc,d5_real_1,d5_subset_1,d5_tarski,d6_nagata_1,d7_nagata_1,d8_binop_1,d8_nagata_1,d9_nagata_1,dt_g1_pre_topc,dt_k10_relset_1,dt_k10_seqfunc,dt_k11_seq_1,dt_k12_mcart_1,dt_k13_finseq_1,dt_k16_complex1,dt_k17_complex1,dt_k18_complex1,dt_k1_binop_1,dt_k1_cantor_1,dt_k1_card_1,dt_k1_domain_1,dt_k1_finseq_1,dt_k1_fraenkel,dt_k1_funct_1,dt_k1_funct_2,dt_k1_metric_1,dt_k1_nagata_1,dt_k1_nagata_2,dt_k1_nat_1,dt_k1_numbers,dt_k1_pcomps_1,dt_k1_prepower,dt_k1_prob_1,dt_k1_real_1,dt_k1_relat_1,dt_k1_seqfunc,dt_k1_square_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_binop_1,dt_k2_finseq_1,dt_k2_finsop_1,dt_k2_funct_1,dt_k2_nagata_1,dt_k2_nagata_2,dt_k2_prepower,dt_k2_prob_1,dt_k2_relat_1,dt_k2_seq_1,dt_k2_series_1,dt_k2_tarski,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_binop_1,dt_k3_cantor_1,dt_k3_card_3,dt_k3_finseq_1,dt_k3_nagata_1,dt_k3_nagata_2,dt_k3_pcomps_1,dt_k3_real_1,dt_k3_seq_4,dt_k3_square_1,dt_k3_subset_1,dt_k3_tarski,dt_k3_topmetr,dt_k3_xboole_0,dt_k4_finseq_1,dt_k4_nagata_1,dt_k4_pscomp_1,dt_k4_relset_1,dt_k4_tarski,dt_k4_xboole_0,dt_k4_xcmplx_0,dt_k5_finsop_1,dt_k5_finsub_1,dt_k5_nagata_1,dt_k5_numbers,dt_k5_ordinal2,dt_k5_real_1,dt_k5_relset_1,dt_k5_seq_1,dt_k5_setfam_1,dt_k5_urysohn3,dt_k6_borsuk_1,dt_k6_pre_topc,dt_k6_real_1,dt_k6_subset_1,dt_k6_xcmplx_0,dt_k7_borsuk_1,dt_k7_urysohn3,dt_k7_xcmplx_0,dt_k8_borsuk_1,dt_k8_funct_2,dt_k9_relat_1,dt_k9_urysohn1,dt_k9_urysohn3,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_cantor_1,dt_m1_finseq_1,dt_m1_fraenkel,dt_m1_relset_1,dt_m1_seqfunc,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_fraenkel,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_cantor_1,existence_m1_finseq_1,existence_m1_fraenkel,existence_m1_relset_1,existence_m1_seqfunc,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_fraenkel,existence_m2_relset_1,existence_m2_subset_1,fc10_finset_1,fc10_membered,fc10_xreal_0,fc11_finseq_1,fc11_finset_1,fc11_membered,fc11_tops_1,fc11_xreal_0,fc12_finset_1,fc12_membered,fc12_xreal_0,fc13_finset_1,fc13_membered,fc13_xreal_0,fc14_finset_1,fc14_membered,fc14_xreal_0,fc15_membered,fc15_xreal_0,fc16_finseq_1,fc16_membered,fc16_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finseq_1,fc1_finset_1,fc1_fraenkel,fc1_nat_1,fc1_pscomp_1,fc1_seq_1,fc1_struct_0,fc1_subset_1,fc1_xreal_0,fc20_xreal_0,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc27_membered,fc27_xreal_0,fc28_membered,fc28_xreal_0,fc29_membered,fc29_xreal_0,fc2_borsuk_1,fc2_finseq_1,fc2_finset_1,fc2_fraenkel,fc2_membered,fc2_pscomp_1,fc2_subset_1,fc2_tops_1,fc30_membered,fc30_xreal_0,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc37_membered,fc38_membered,fc39_membered,fc3_nat_1,fc3_pscomp_1,fc3_subset_1,fc3_tops_1,fc3_xreal_0,fc40_membered,fc41_membered,fc4_nat_1,fc4_subset_1,fc4_tops_1,fc5_membered,fc5_seq_1,fc5_xreal_0,fc6_membered,fc6_xreal_0,fc7_membered,fc7_topreal6,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_finseq_1,fc9_finset_1,fc9_membered,fc9_xreal_0,fraenkel_a_0_42_fraenkel,fraenkel_a_2_1_borsuk_1,fraenkel_a_3_0_borsuk_1,fraenkel_a_3_0_pcomps_1,fraenkel_a_4_0_nagata_2,fraenkel_a_4_1_nagata_2,fraenkel_a_4_2_nagata_2,fraenkel_a_4_6_nagata_2,fraenkel_a_5_0_nagata_2,fraenkel_a_6_0_nagata_2,fraenkel_a_6_1_nagata_2,fraenkel_a_6_2_nagata_2,fraenkel_a_6_7_nagata_2,fraenkel_a_6_8_nagata_2,fraenkel_a_7_0_nagata_2,fraenkel_a_8_0_nagata_2,fraenkel_a_8_1_nagata_2,free_g1_pre_topc,idempotence_k1_square_1,idempotence_k2_xboole_0,idempotence_k3_square_1,idempotence_k3_xboole_0,involutiveness_k1_real_1,involutiveness_k3_subset_1,involutiveness_k4_xcmplx_0,l8_nagata_2,projectivity_k16_complex1,projectivity_k17_complex1,projectivity_k18_complex1,projectivity_k1_prepower,rc1_finseq_1,rc1_finset_1,rc1_fraenkel,rc1_membered,rc1_nat_1,rc1_pscomp_1,rc1_seq_1,rc1_subset_1,rc1_tops_1,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_nat_1,rc2_pscomp_1,rc2_subset_1,rc2_tops_1,rc2_xreal_0,rc3_finseq_1,rc3_finset_1,rc3_nat_1,rc3_pscomp_1,rc3_struct_0,rc3_tops_1,rc3_xreal_0,rc4_finseq_1,rc4_finset_1,rc4_pscomp_1,rc4_tops_1,rc4_xreal_0,rc5_struct_0,rc5_tops_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rc9_pscomp_1,redefinition_k10_relset_1,redefinition_k11_seq_1,redefinition_k12_mcart_1,redefinition_k17_complex1,redefinition_k18_complex1,redefinition_k1_domain_1,redefinition_k1_fraenkel,redefinition_k1_metric_1,redefinition_k1_nagata_1,redefinition_k1_nat_1,redefinition_k1_pcomps_1,redefinition_k1_prepower,redefinition_k1_prob_1,redefinition_k1_real_1,redefinition_k1_seqfunc,redefinition_k2_binop_1,redefinition_k2_finseq_1,redefinition_k2_nagata_1,redefinition_k2_prob_1,redefinition_k2_seq_1,redefinition_k3_cantor_1,redefinition_k3_finseq_1,redefinition_k3_real_1,redefinition_k3_square_1,redefinition_k4_finseq_1,redefinition_k4_pscomp_1,redefinition_k4_relset_1,redefinition_k5_finsop_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k5_relset_1,redefinition_k5_setfam_1,redefinition_k5_urysohn3,redefinition_k6_real_1,redefinition_k6_subset_1,redefinition_k7_borsuk_1,redefinition_k7_urysohn3,redefinition_k8_borsuk_1,redefinition_k8_funct_2,redefinition_k9_urysohn1,redefinition_k9_urysohn3,redefinition_m2_finseq_1,redefinition_m2_fraenkel,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_binop_1__e22_23_1_1__nagata_2,s1_funct_2__e26_23_1_1__nagata_2,s1_funct_2__e32_23_1_1__nagata_2,s1_funct_2__e7_23_1_1__nagata_2,s1_funct_2__e9_23_1__nagata_2,s1_nat_1__e31_23_1_1_19__nagata_2,s1_nat_1__e7_23_1_1_17_2_1_1_2__nagata_2,s1_nat_1__e7_23_1_1_17_4_1_2__nagata_2,s1_nat_1__e7_23_1_1_18_1_1_1_2__nagata_2,s1_nat_1__e7_23_1_1_18_2_1_1_2__nagata_2,s21_fraenkel__e2_23_1_1_4__nagata_2,s21_fraenkel__e4_23_1_1_7__nagata_2,s2_funct_2__e3_23_1_1__nagata_2,s3_funct_2__e10_23_1_1__nagata_2,s3_funct_2__e14_23_1_1__nagata_2,s3_funct_2__e19_23_1_1__nagata_2,s7_funct_2__e2_23_1_1_5__nagata_2,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r1_xboole_0,t106_zfmisc_1,t10_xreal_1,t11_binop_1,t11_funct_2,t121_funct_2,t12_pcomps_1,t132_complex1,t138_complex1,t13_finset_1,t142_complex1,t14_finset_1,t15_finset_1,t15_nagata_1,t16_nagata_1,t17_nagata_2,t17_xboole_1,t18_nagata_2,t19_nagata_1,t1_arithm,t1_boole,t1_numerals,t1_real,t1_seqfunc,t1_subset,t1_xboole_1,t20_nagata_1,t21_nagata_1,t23_nagata_1,t23_pcomps_1,t23_urysohn3,t24_topmetr,t25_finseq_1,t25_nagata_1,t25_prob_1,t26_nagata_1,t26_subset_1,t27_finseq_1,t27_finseq_3,t27_nagata_1,t28_nagata_1,t28_xboole_1,t29_tops_1,t2_boole,t2_nagata_2,t2_real,t2_subset,t2_tarski,t2_xreal_1,t30_nagata_1,t33_zfmisc_1,t35_square_1,t37_xboole_1,t38_nat_1,t39_nat_1,t3_boole,t3_finseq_1,t3_real,t3_subset,t3_xboole_0,t43_subset_1,t46_borsuk_1,t47_xboole_1,t48_pre_topc,t4_arithm,t4_boole,t4_real,t4_subset,t54_funct_1,t5_arithm,t5_real,t5_subset,t60_seq_4,t63_xboole_1,t6_arithm,t6_boole,t6_real,t73_finseq_4,t7_absvalue,t7_boole,t7_funct_2,t7_real,t83_topreal6,t8_boole,t8_real,t92_zfmisc_1,t9_xreal_1]), [file(nagata_2,t19_nagata_2)]). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(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(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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_finset_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), []). fof(cc1_fraenkel,axiom,( ! [A] : ( v1_fraenkel(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) ) ) ) ), file(fraenkel,cc1_fraenkel), []). 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_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_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_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(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_finset_1,axiom,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_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_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(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_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(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). 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(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(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_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(cc7_xreal_0,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), []). fof(cc9_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), []). fof(commutativity_k11_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k11_seq_1(A,B) = k11_seq_1(B,A) ) ), file(seq_1,k11_seq_1), []). 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_k1_square_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => k1_square_1(A,B) = k1_square_1(B,A) ) ), file(square_1,k1_square_1), []). fof(commutativity_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(commutativity_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), []). fof(commutativity_k3_square_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_square_1(A,B) = k3_square_1(B,A) ) ), file(square_1,k3_square_1), []). 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_k5_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => k5_seq_1(A,B) = k5_seq_1(B,A) ) ), file(seq_1,k5_seq_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(d11_finseq_1,axiom,( ! [A,B] : ( B = k13_finseq_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> m2_finseq_1(C,A) ) ) ), file(finseq_1,d11_finseq_1), []). fof(d12_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k9_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( r2_hidden(E,k1_relat_1(A)) & r2_hidden(E,B) & D = k1_funct_1(A,E) ) ) ) ) ), file(funct_1,d12_funct_1), []). fof(d1_absvalue,axiom,( ! [A] : ( v1_xreal_0(A) => ( ( r1_xreal_0(0,A) => k16_complex1(A) = A ) & ( ~ r1_xreal_0(0,A) => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) ), file(absvalue,d1_absvalue), []). fof(d1_binop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : k1_binop_1(A,B,C) = k1_funct_1(A,k4_tarski(B,C)) ) ), file(binop_1,d1_binop_1), []). fof(d1_finsop_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m2_relset_1(C,k2_zfmisc_1(A,A),A) ) => ( ( v1_setwiseo(C,A) | r1_xreal_0(1,k3_finseq_1(B)) ) => ! [D] : ( m1_subset_1(D,A) => ( ( ( v1_setwiseo(C,A) & k3_finseq_1(B) = 0 ) => ( D = k2_finsop_1(A,B,C) <=> D = k3_binop_1(A,C) ) ) & ( ~ ( v1_setwiseo(C,A) & k3_finseq_1(B) = 0 ) => ( D = k2_finsop_1(A,B,C) <=> ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k5_numbers,A) & m2_relset_1(E,k5_numbers,A) & k8_funct_2(k5_numbers,A,E,1) = k1_funct_1(B,1) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ~ ( 0 != F & ~ r1_xreal_0(k3_finseq_1(B),F) & k8_funct_2(k5_numbers,A,E,k1_nat_1(F,1)) != k1_binop_1(C,k8_funct_2(k5_numbers,A,E,F),k1_funct_1(B,k1_nat_1(F,1))) ) ) & D = k8_funct_2(k5_numbers,A,E,k3_finseq_1(B)) ) ) ) ) ) ) ) ) ) ), file(finsop_1,d1_finsop_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_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) => k1_square_1(A,B) = A ) & ( ~ r1_xreal_0(A,B) => k1_square_1(A,B) = B ) ) ) ) ), file(square_1,d1_square_1), []). fof(d1_xboole_0,axiom,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), []). fof(d2_cantor_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( m1_cantor_1(B,A) <=> ( r1_tarski(B,u1_pre_topc(A)) & r1_tarski(u1_pre_topc(A),k1_cantor_1(u1_struct_0(A),B)) ) ) ) ) ), file(cantor_1,d2_cantor_1), []). fof(d2_nagata_2,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,k1_numbers) & m2_relset_1(D,A,k1_numbers) ) => ( D = k2_nagata_2(A,B,C) <=> ! [E] : ( m1_subset_1(E,A) => k2_seq_1(A,k1_numbers,D,E) = k1_metric_1(A,A,B,C,E) ) ) ) ) ) ), file(nagata_2,d2_nagata_2), []). fof(d2_pcomps_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_pcomps_1(B,A) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & r2_hidden(C,D) & v3_pre_topc(D,A) & v1_finset_1(a_3_0_pcomps_1(A,B,D)) ) ) ) ) ) ), file(pcomps_1,d2_pcomps_1), []). fof(d2_setwiseo,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m2_relset_1(B,k2_zfmisc_1(A,A),A) ) => ( v1_setwiseo(B,A) <=> ? [C] : ( m1_subset_1(C,A) & r3_binop_1(A,C,B) ) ) ) ) ), file(setwiseo,d2_setwiseo), []). 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(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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k3_finseq_1(A) <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ), file(finseq_1,d3_finseq_1), []). fof(d3_funct_2,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ( v2_funct_2(C,A,B) <=> k2_relat_1(C) = B ) ) ), file(funct_2,d3_funct_2), []). fof(d3_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => ( v3_nagata_1(B,A) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => v1_pcomps_1(k1_nagata_1(A,B,C),A) ) ) ) ) ), file(nagata_1,d3_nagata_1), []). fof(d3_nagata_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,k1_numbers) & m2_relset_1(D,A,k1_numbers) ) => ( D = k3_nagata_2(A,B,C) <=> ! [E] : ( m1_subset_1(E,A) => k8_funct_2(A,k1_numbers,D,E) = k4_pscomp_1(k10_relset_1(A,k1_numbers,k2_nagata_2(A,B,E),C)) ) ) ) ) ) ) ), file(nagata_2,d3_nagata_2), []). fof(d3_pcomps_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( C = k3_pcomps_1(A,B) <=> ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(D,C) <=> ? [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) & D = k6_pre_topc(A,E) & r2_hidden(E,B) ) ) ) ) ) ) ) ), file(pcomps_1,d3_pcomps_1), []). 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_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( m1_finseq_1(B,A) <=> r1_tarski(k2_relat_1(B),A) ) ) ), file(finseq_1,d4_finseq_1), []). fof(d4_funct_2,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ( v3_funct_2(C,A,B) <=> ( v2_funct_1(C) & v2_funct_2(C,A,B) ) ) ) ), file(funct_2,d4_funct_2), []). 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(d4_xboole_0,axiom,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), []). 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_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_pre_topc(B,A) <=> r2_hidden(B,u1_pre_topc(A)) ) ) ) ), file(pre_topc,d5_pre_topc), []). fof(d5_real_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), []). fof(d5_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,B) = k4_xboole_0(A,B) ) ), file(subset_1,d5_subset_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(d6_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => ( v6_nagata_1(B,A) <=> ( v3_nagata_1(B,A) & m1_cantor_1(k2_nagata_1(A,B),A) ) ) ) ) ), file(nagata_1,d6_nagata_1), []). fof(d7_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m2_relset_1(B,u1_struct_0(A),k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m2_relset_1(C,u1_struct_0(A),k1_numbers) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_numbers) & m2_relset_1(D,u1_struct_0(A),k1_numbers) ) => ( D = k3_nagata_1(A,B,C) <=> ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => k8_funct_2(u1_struct_0(A),k1_numbers,D,E) = k3_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,E),k8_funct_2(u1_struct_0(A),k1_numbers,C,E)) ) ) ) ) ) ) ), file(nagata_1,d7_nagata_1), []). fof(d8_binop_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m2_relset_1(B,k2_zfmisc_1(A,A),A) ) => ( ? [C] : ( m1_subset_1(C,A) & r3_binop_1(A,C,B) ) => ! [C] : ( m1_subset_1(C,A) => ( C = k3_binop_1(A,B) <=> r3_binop_1(A,C,B) ) ) ) ) ), file(binop_1,d8_binop_1), []). fof(d8_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) & m2_relset_1(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(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 = k4_nagata_1(A,B,C,D) <=> ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => k7_urysohn3(A,B,E,F) = k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),D,F),F) ) ) ) ) ) ) ) ), file(nagata_1,d8_nagata_1), []). fof(d9_nagata_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,k1_numbers) & m2_relset_1(C,A,k1_numbers) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,k1_numbers) & m2_relset_1(D,A,k1_numbers) ) => ( D = k5_nagata_1(A,B,C) <=> ! [E] : ( r2_hidden(E,A) => k1_funct_1(D,E) = k1_square_1(B,k1_funct_1(C,E)) ) ) ) ) ) ), file(nagata_1,d9_nagata_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_relset_1,axiom,( ! [A,B,C,D] : ( m1_relset_1(C,A,B) => m1_subset_1(k10_relset_1(A,B,C,D),k1_zfmisc_1(B)) ) ), file(relset_1,k10_relset_1), []). fof(dt_k10_seqfunc,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_seqfunc(B,A,k1_numbers) & m1_subset_1(C,A) ) => ( v1_funct_1(k10_seqfunc(A,B,C)) & v1_funct_2(k10_seqfunc(A,B,C),k5_numbers,k1_numbers) & m2_relset_1(k10_seqfunc(A,B,C),k5_numbers,k1_numbers) ) ) ), file(seqfunc,k10_seqfunc), []). fof(dt_k11_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_funct_1(k11_seq_1(A,B)) & v1_funct_2(k11_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k11_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k11_seq_1), []). 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_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), []). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_complex1), []). fof(dt_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k17_complex1(A),k1_numbers) ) ), file(complex1,k17_complex1), []). fof(dt_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => m1_subset_1(k18_complex1(A),k1_numbers) ) ), file(complex1,k18_complex1), []). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k1_cantor_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => m1_subset_1(k1_cantor_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ), file(cantor_1,k1_cantor_1), []). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), []). fof(dt_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) ), file(domain_1,k1_domain_1), []). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), []). fof(dt_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => m1_fraenkel(k1_fraenkel(A,B),A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_funct_2,axiom,( $true ), file(funct_2,k1_funct_2), []). fof(dt_k1_metric_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(C) & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers) & m1_subset_1(D,A) & m1_subset_1(E,B) ) => m1_subset_1(k1_metric_1(A,B,C,D,E),k1_numbers) ) ), file(metric_1,k1_metric_1), []). fof(dt_k1_nagata_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k1_nagata_1(A,B,C),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(nagata_1,k1_nagata_1), []). fof(dt_k1_nagata_2,axiom, ( v1_funct_1(k1_nagata_2) & v1_funct_2(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) & m2_relset_1(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) ), file(nagata_2,k1_nagata_2), []). 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_prepower,axiom,( ! [A] : ( v1_int_1(A) => m2_subset_1(k1_prepower(A),k1_numbers,k5_numbers) ) ), file(prepower,k1_prepower), []). fof(dt_k1_prob_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A)) & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k1_prob_1(A,B,C),k1_zfmisc_1(A)) ) ), file(prob_1,k1_prob_1), []). fof(dt_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k1_real_1(A),k1_numbers) ) ), file(real_1,k1_real_1), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_seqfunc,axiom,( ! [A,B,C,D] : ( ( m1_seqfunc(C,A,B) & v4_ordinal2(D) ) => ( v1_funct_1(k1_seqfunc(A,B,C,D)) & m2_relset_1(k1_seqfunc(A,B,C,D),A,B) ) ) ), file(seqfunc,k1_seqfunc), []). fof(dt_k1_square_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k1_square_1(A,B)) ) ), file(square_1,k1_square_1), []). 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_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => m1_subset_1(k2_binop_1(A,B,C,D,E,F),C) ) ), file(binop_1,k2_binop_1), []). fof(dt_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_1), []). fof(dt_k2_finsop_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_finseq_1(B,A) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(A,A),A) & m1_relset_1(C,k2_zfmisc_1(A,A),A) ) => m1_subset_1(k2_finsop_1(A,B,C),A) ) ), file(finsop_1,k2_finsop_1), []). fof(dt_k2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_relat_1(k2_funct_1(A)) & v1_funct_1(k2_funct_1(A)) ) ) ), file(funct_1,k2_funct_1), []). fof(dt_k2_nagata_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => m1_subset_1(k2_nagata_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(nagata_1,k2_nagata_1), []). fof(dt_k2_nagata_2,axiom,( ! [A,B,C] : ( ( 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) & m1_subset_1(C,A) ) => ( v1_funct_1(k2_nagata_2(A,B,C)) & v1_funct_2(k2_nagata_2(A,B,C),A,k1_numbers) & m2_relset_1(k2_nagata_2(A,B,C),A,k1_numbers) ) ) ), file(nagata_2,k2_nagata_2), []). fof(dt_k2_prepower,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_funct_1(k2_prepower(A)) & v1_funct_2(k2_prepower(A),k5_numbers,k1_numbers) & m2_relset_1(k2_prepower(A),k5_numbers,k1_numbers) ) ) ), file(prepower,k2_prepower), []). fof(dt_k2_prob_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A)) & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) ) => m1_subset_1(k2_prob_1(A,B),k1_zfmisc_1(A)) ) ), file(prob_1,k2_prob_1), []). 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_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => m1_subset_1(k2_series_1(A),k1_numbers) ) ), file(series_1,k2_series_1), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), []). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), []). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), []). fof(dt_k3_binop_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) ) => m1_subset_1(k3_binop_1(A,B),A) ) ), file(binop_1,k3_binop_1), []). fof(dt_k3_cantor_1,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) & v1_funct_1(D) & v1_funct_2(D,B,k1_zfmisc_1(C)) & m1_relset_1(D,B,k1_zfmisc_1(C)) ) => m1_subset_1(k3_cantor_1(A,B,C,D,E),k1_zfmisc_1(k1_zfmisc_1(A))) ) ), file(cantor_1,k3_cantor_1), []). fof(dt_k3_card_3,axiom,( $true ), file(card_3,k3_card_3), []). 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_nagata_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m1_relset_1(B,u1_struct_0(A),k1_numbers) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m1_relset_1(C,u1_struct_0(A),k1_numbers) ) => ( v1_funct_1(k3_nagata_1(A,B,C)) & v1_funct_2(k3_nagata_1(A,B,C),u1_struct_0(A),k1_numbers) & m2_relset_1(k3_nagata_1(A,B,C),u1_struct_0(A),k1_numbers) ) ) ), file(nagata_1,k3_nagata_1), []). fof(dt_k3_nagata_2,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & 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) & m1_subset_1(C,k1_zfmisc_1(A)) ) => ( v1_funct_1(k3_nagata_2(A,B,C)) & v1_funct_2(k3_nagata_2(A,B,C),A,k1_numbers) & m2_relset_1(k3_nagata_2(A,B,C),A,k1_numbers) ) ) ), file(nagata_2,k3_nagata_2), []). fof(dt_k3_pcomps_1,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) => m1_subset_1(k3_pcomps_1(A,B),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pcomps_1,k3_pcomps_1), []). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), []). fof(dt_k3_seq_4,axiom,( ! [A] : ( v2_membered(A) => v1_xreal_0(k3_seq_4(A)) ) ), file(seq_4,k3_seq_4), []). fof(dt_k3_square_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_square_1(A,B),k1_numbers) ) ), file(square_1,k3_square_1), []). 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_topmetr,axiom, ( v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & l1_pre_topc(k3_topmetr) ), file(topmetr,k3_topmetr), []). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), []). 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_nagata_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) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m1_relset_1(C,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) & m1_relset_1(D,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) ) => ( v1_funct_1(k4_nagata_1(A,B,C,D)) & v1_funct_2(k4_nagata_1(A,B,C,D),u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(k4_nagata_1(A,B,C,D),u1_struct_0(A),u1_struct_0(B)) ) ) ), file(nagata_1,k4_nagata_1), []). 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_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_finsop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k5_finsop_1(A),k5_finsub_1(k5_numbers)) ) ), file(finsop_1,k5_finsop_1), []). fof(dt_k5_finsub_1,axiom,( ! [A] : v4_finsub_1(k5_finsub_1(A)) ), file(finsub_1,k5_finsub_1), []). fof(dt_k5_nagata_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_numbers) & v1_funct_1(C) & v1_funct_2(C,A,k1_numbers) & m1_relset_1(C,A,k1_numbers) ) => ( v1_funct_1(k5_nagata_1(A,B,C)) & v1_funct_2(k5_nagata_1(A,B,C),A,k1_numbers) & m2_relset_1(k5_nagata_1(A,B,C),A,k1_numbers) ) ) ), file(nagata_1,k5_nagata_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_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), []). fof(dt_k5_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_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k5_seq_1(A,B)) & v1_funct_1(k5_seq_1(A,B)) ) ) ), file(seq_1,k5_seq_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_urysohn3,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,A,k1_pcomps_1(u1_struct_0(B))) & m1_relset_1(C,A,k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(D,A) ) => m1_subset_1(k5_urysohn3(A,B,C,D),k1_zfmisc_1(u1_struct_0(B))) ) ), file(urysohn3,k5_urysohn3), []). 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_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_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), []). fof(dt_k6_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_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). 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_k7_urysohn3,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) & 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(D,u1_struct_0(A)) ) => m1_subset_1(k7_urysohn3(A,B,C,D),u1_struct_0(B)) ) ), file(urysohn3,k7_urysohn3), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). 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_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_1), []). fof(dt_k9_urysohn1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k9_urysohn1(A,B,C),k1_numbers) ) ), file(urysohn1,k9_urysohn1), []). fof(dt_k9_urysohn3,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k9_urysohn3(A,B,C),k1_numbers) ) ), file(urysohn3,k9_urysohn3), []). 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_cantor_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_cantor_1(B,A) => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) ), file(cantor_1,m1_cantor_1), []). 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_fraenkel,axiom,( ! [A,B,C] : ( m1_fraenkel(C,A,B) => ( ~ v1_xboole_0(C) & v1_fraenkel(C) ) ) ), file(fraenkel,m1_fraenkel), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_seqfunc,axiom,( ! [A,B,C] : ( m1_seqfunc(C,A,B) => ( v1_relat_1(C) & v1_funct_1(C) ) ) ), file(seqfunc,m1_seqfunc), []). 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_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ! [D] : ( m2_fraenkel(D,A,B,C) => ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) ) ) ), file(fraenkel,m2_fraenkel), []). 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_cantor_1,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_cantor_1(B,A) ) ), file(cantor_1,m1_cantor_1), []). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), []). fof(existence_m1_fraenkel,axiom,( ! [A,B] : ? [C] : m1_fraenkel(C,A,B) ), file(fraenkel,m1_fraenkel), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_seqfunc,axiom,( ! [A,B] : ? [C] : m1_seqfunc(C,A,B) ), file(seqfunc,m1_seqfunc), []). 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_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ? [D] : m2_fraenkel(D,A,B,C) ) ), file(fraenkel,m2_fraenkel), []). 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_finset_1,axiom,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_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_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k2_relat_1(A)) ) ), file(finseq_1,fc11_finseq_1), []). fof(fc11_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), []). 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(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_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), []). 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_finset_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(B) ) => v1_finset_1(k9_relat_1(A,B)) ) ), file(finset_1,fc13_finset_1), []). fof(fc13_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(fc13_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), []). fof(fc14_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), []). fof(fc14_membered,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), []). fof(fc14_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), []). fof(fc15_membered,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), []). fof(fc15_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), []). fof(fc16_finseq_1,axiom,( ! [A] : ( ~ v1_xboole_0(k13_finseq_1(A)) & v1_fraenkel(k13_finseq_1(A)) ) ), file(finseq_1,fc16_finseq_1), []). fof(fc16_membered,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), []). fof(fc16_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), []). fof(fc17_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_1), []). fof(fc17_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), []). fof(fc18_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), []). fof(fc19_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), []). fof(fc1_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(k1_finseq_1(A)) ) ), file(finseq_1,fc1_finseq_1), []). fof(fc1_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_fraenkel,axiom,( ! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ), file(fraenkel,fc1_fraenkel), []). 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_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_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), []). fof(fc20_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), []). fof(fc22_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). fof(fc23_membered,axiom,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), []). fof(fc24_membered,axiom,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), []). fof(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc26_membered,axiom,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), []). fof(fc27_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), []). fof(fc27_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), []). fof(fc28_membered,axiom,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), []). fof(fc28_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), []). fof(fc29_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), []). fof(fc29_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), []). fof(fc2_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_finseq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), []). fof(fc2_finset_1,axiom,( ! [A,B] : ( ~ v1_xboole_0(k2_tarski(A,B)) & v1_finset_1(k2_tarski(A,B)) ) ), file(finset_1,fc2_finset_1), []). fof(fc2_fraenkel,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => ( v1_finset_1(k1_funct_2(A,B)) & v1_fraenkel(k1_funct_2(A,B)) ) ) ), file(fraenkel,fc2_fraenkel), []). 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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). 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(fc30_membered,axiom,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), []). fof(fc30_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), []). fof(fc31_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), []). fof(fc32_membered,axiom,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), []). fof(fc33_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), []). fof(fc34_membered,axiom,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), []). fof(fc35_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), []). fof(fc36_membered,axiom,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), []). fof(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_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_pscomp_1,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => ~ v1_xboole_0(k9_relat_1(C,A)) ) ), file(pscomp_1,fc3_pscomp_1), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_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(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(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_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_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_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_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_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k5_seq_1(A,B)) & v1_funct_1(k5_seq_1(A,B)) & v1_seq_1(k5_seq_1(A,B)) ) ) ), file(seq_1,fc5_seq_1), []). fof(fc5_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc6_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), []). fof(fc7_membered,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), []). 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_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_finseq_1,axiom,( ! [A] : ~ v1_xboole_0(k13_finseq_1(A)) ), file(finseq_1,fc9_finseq_1), []). fof(fc9_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), []). 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_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), []). fof(fraenkel_a_0_42_fraenkel,axiom,( ! [A] : ( r2_hidden(A,a_0_42_fraenkel) <=> ? [B] : ( m1_subset_1(B,f1_s21_fraenkel) & A = f3_s21_fraenkel(B) & r2_hidden(B,f2_s21_fraenkel) ) ) ), file(fraenkel,a_0_42_fraenkel), []). 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_0_pcomps_1,axiom,( ! [A,B,C,D] : ( ( l1_pre_topc(B) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) ) => ( r2_hidden(A,a_3_0_pcomps_1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) & A = E & r2_hidden(E,C) & ~ r1_xboole_0(E,D) ) ) ) ), file(pcomps_1,a_3_0_pcomps_1), []). fof(fraenkel_a_4_0_nagata_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_4_0_nagata_2(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B))) & A = F & r2_hidden(F,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & r1_tarski(k6_pre_topc(B,F),E) ) ) ) ), file(nagata_2,a_4_0_nagata_2), []). fof(fraenkel_a_4_1_nagata_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) ) => ( r2_hidden(A,a_4_1_nagata_2(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B))) & A = F & r2_hidden(F,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & r1_tarski(k6_pre_topc(B,F),E) ) ) ) ), file(nagata_2,a_4_1_nagata_2), []). fof(fraenkel_a_4_2_nagata_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & m2_subset_1(E,k1_zfmisc_1(u1_struct_0(B)),k1_pcomps_1(u1_struct_0(B))) ) => ( r2_hidden(A,a_4_2_nagata_2(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B))) & A = F & r2_hidden(F,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & ~ r1_xboole_0(F,E) ) ) ) ), file(nagata_2,a_4_2_nagata_2), []). fof(fraenkel_a_4_6_nagata_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & m1_subset_1(E,k1_pcomps_1(u1_struct_0(B))) ) => ( r2_hidden(A,a_4_6_nagata_2(B,C,D,E)) <=> ? [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B))) & A = F & r2_hidden(F,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & ~ r1_xboole_0(F,E) ) ) ) ), file(nagata_2,a_4_6_nagata_2), []). fof(fraenkel_a_5_0_nagata_2,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(E,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(F,u1_struct_0(B)) ) => ( r2_hidden(A,a_5_0_nagata_2(B,C,D,E,F)) <=> ? [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B))) & A = G & r2_hidden(G,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & ~ r1_xboole_0(G,k8_funct_2(u1_struct_0(B),k1_pcomps_1(u1_struct_0(B)),E,F)) ) ) ) ), file(nagata_2,a_5_0_nagata_2), []). fof(fraenkel_a_6_0_nagata_2,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(G,u1_struct_0(B)) ) => ( r2_hidden(A,a_6_0_nagata_2(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) & A = k1_funct_1(E,H) & r2_hidden(H,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & r2_hidden(H,a_5_0_nagata_2(B,C,D,F,G)) ) ) ) ), file(nagata_2,a_6_0_nagata_2), []). fof(fraenkel_a_6_1_nagata_2,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(G,u1_struct_0(B)) ) => ( r2_hidden(A,a_6_1_nagata_2(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) & A = k1_funct_1(E,H) & r2_hidden(H,a_5_0_nagata_2(B,C,D,F,G)) ) ) ) ), file(nagata_2,a_6_1_nagata_2), []). fof(fraenkel_a_6_2_nagata_2,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m1_subset_1(F,u1_struct_0(B)) & m1_subset_1(G,u1_struct_0(B)) ) => ( r2_hidden(A,a_6_2_nagata_2(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) & A = k1_funct_1(E,H) & r2_hidden(H,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,u1_struct_0(B),k1_numbers) & m2_relset_1(I,u1_struct_0(B),k1_numbers) ) => ~ ( k1_funct_1(E,H) = I & r1_xreal_0(k8_funct_2(u1_struct_0(B),k1_numbers,I,F),0) & r1_xreal_0(k8_funct_2(u1_struct_0(B),k1_numbers,I,G),0) ) ) ) ) ) ), file(nagata_2,a_6_2_nagata_2), []). fof(fraenkel_a_6_7_nagata_2,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(G,u1_struct_0(B)) ) => ( r2_hidden(A,a_6_7_nagata_2(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) & A = k1_funct_1(E,H) & r2_hidden(H,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & r2_hidden(H,a_6_8_nagata_2(B,C,D,F,G,H)) ) ) ) ), file(nagata_2,a_6_7_nagata_2), []). fof(fraenkel_a_6_8_nagata_2,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(E,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(F,u1_struct_0(B)) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B))) ) => ( r2_hidden(A,a_6_8_nagata_2(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(B))) & A = G & r2_hidden(G,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D)) & ~ r1_xboole_0(G,k8_funct_2(u1_struct_0(B),k1_pcomps_1(u1_struct_0(B)),E,F)) ) ) ) ), file(nagata_2,a_6_8_nagata_2), []). fof(fraenkel_a_7_0_nagata_2,axiom,( ! [A,B,C,D,E,F,G,H] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,k1_fraenkel(u1_struct_0(B),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(B,B)),k1_numbers)) & m2_relset_1(F,k1_fraenkel(u1_struct_0(B),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(B,B)),k1_numbers)) & m1_subset_1(G,u1_struct_0(B)) & m1_subset_1(H,u1_struct_0(B)) ) => ( r2_hidden(A,a_7_0_nagata_2(B,C,D,E,F,G,H)) <=> ? [I] : ( v1_funct_1(I) & v1_funct_2(I,u1_struct_0(B),k1_numbers) & m2_relset_1(I,u1_struct_0(B),k1_numbers) & A = k1_funct_1(F,I) & r2_hidden(I,a_6_2_nagata_2(B,C,D,E,G,H)) ) ) ) ), file(nagata_2,a_7_0_nagata_2), []). fof(fraenkel_a_8_0_nagata_2,axiom,( ! [A,B,C,D,E,F,G,H,I] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & v1_funct_1(G) & v1_funct_2(G,k1_fraenkel(u1_struct_0(B),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(B,B)),k1_numbers)) & m2_relset_1(G,k1_fraenkel(u1_struct_0(B),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(B,B)),k1_numbers)) & m1_subset_1(H,u1_struct_0(B)) & m1_subset_1(I,u1_struct_0(B)) ) => ( r2_hidden(A,a_8_0_nagata_2(B,C,D,E,F,G,H,I)) <=> ? [J] : ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(B),k1_numbers) & m2_relset_1(J,u1_struct_0(B),k1_numbers) & A = k1_funct_1(G,J) & r2_hidden(J,k2_xboole_0(a_6_0_nagata_2(B,C,D,E,F,H),a_6_0_nagata_2(B,C,D,E,F,I))) ) ) ) ), file(nagata_2,a_8_0_nagata_2), []). fof(fraenkel_a_8_1_nagata_2,axiom,( ! [A,B,C,D,E,F,G,H,I] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_relset_1(C,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(B)))) & m2_subset_1(D,k1_numbers,k5_numbers) & v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(B),k5_numbers,k1_pcomps_1(u1_struct_0(B)),C,D),k1_fraenkel(u1_struct_0(B),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & m2_relset_1(F,u1_struct_0(B),k1_pcomps_1(u1_struct_0(B))) & v1_funct_1(G) & v1_funct_2(G,k1_fraenkel(u1_struct_0(B),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(B,B)),k1_numbers)) & m2_relset_1(G,k1_fraenkel(u1_struct_0(B),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(B,B)),k1_numbers)) & m1_subset_1(H,u1_struct_0(B)) & m1_subset_1(I,u1_struct_0(B)) ) => ( r2_hidden(A,a_8_1_nagata_2(B,C,D,E,F,G,H,I)) <=> ? [J] : ( m2_fraenkel(J,u1_struct_0(B),k1_numbers,k1_fraenkel(u1_struct_0(B),k1_numbers)) & A = k1_funct_1(G,J) & r2_hidden(J,k2_xboole_0(a_6_0_nagata_2(B,C,D,E,F,H),a_6_0_nagata_2(B,C,D,E,F,I))) ) ) ) ), file(nagata_2,a_8_1_nagata_2), []). 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_k1_square_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => k1_square_1(A,A) = A ) ), file(square_1,k1_square_1), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k3_square_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_square_1(A,A) = A ) ), file(square_1,k3_square_1), []). fof(idempotence_k3_xboole_0,axiom,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), []). fof(involutiveness_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(k1_real_1(A)) = A ) ), file(real_1,k1_real_1), []). fof(involutiveness_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(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(l8_nagata_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( r1_nagata_1(A,B) => ( v2_seq_2(B) & ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(A)) ) => ! [D] : ( m1_subset_1(D,A) => ( ~ v1_xboole_0(k10_relset_1(A,k1_numbers,k2_nagata_2(A,B,D),C)) & v2_seq_4(k10_relset_1(A,k1_numbers,k2_nagata_2(A,B,D),C)) ) ) ) ) ) ) ) ), file(nagata_2,l8_nagata_2), []). fof(projectivity_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), []). fof(projectivity_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k17_complex1(A)) = k17_complex1(A) ) ), file(complex1,k17_complex1), []). fof(projectivity_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ), file(complex1,k18_complex1), []). fof(projectivity_k1_prepower,axiom,( ! [A] : ( v1_int_1(A) => k1_prepower(k1_prepower(A)) = k1_prepower(A) ) ), file(prepower,k1_prepower), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_finset_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), []). fof(rc1_fraenkel,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_fraenkel(A) ) ), file(fraenkel,rc1_fraenkel), []). 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_pscomp_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) ), file(pscomp_1,rc1_pscomp_1), []). 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_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_finseq_1,axiom,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), []). fof(rc2_finset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_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_pscomp_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_setfam_1(A) ) ), file(pscomp_1,rc2_pscomp_1), []). 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_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_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), []). fof(rc3_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_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_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_finseq_1,axiom,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), []). fof(rc4_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), []). 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_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_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_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(rc7_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(rc9_pscomp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v6_compts_1(B,A) ) ) ), file(pscomp_1,rc9_pscomp_1), []). fof(redefinition_k10_relset_1,axiom,( ! [A,B,C,D] : ( m1_relset_1(C,A,B) => k10_relset_1(A,B,C,D) = k9_relat_1(C,D) ) ), file(relset_1,k10_relset_1), []). fof(redefinition_k11_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k11_seq_1(A,B) = k5_seq_1(A,B) ) ), file(seq_1,k11_seq_1), []). 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_k17_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(A) = k16_complex1(A) ) ), file(complex1,k17_complex1), []). fof(redefinition_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(A) = k16_complex1(A) ) ), file(complex1,k18_complex1), []). fof(redefinition_k1_domain_1,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(C,A) & m1_subset_1(D,B) ) => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) ), file(domain_1,k1_domain_1), []). fof(redefinition_k1_fraenkel,axiom,( ! [A,B] : ( ~ v1_xboole_0(B) => k1_fraenkel(A,B) = k1_funct_2(A,B) ) ), file(fraenkel,k1_fraenkel), []). fof(redefinition_k1_metric_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(C) & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers) & m1_subset_1(D,A) & m1_subset_1(E,B) ) => k1_metric_1(A,B,C,D,E) = k1_binop_1(C,D,E) ) ), file(metric_1,k1_metric_1), []). fof(redefinition_k1_nagata_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m1_subset_1(C,k5_numbers) ) => k1_nagata_1(A,B,C) = k1_funct_1(B,C) ) ), file(nagata_1,k1_nagata_1), []). 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_prepower,axiom,( ! [A] : ( v1_int_1(A) => k1_prepower(A) = k16_complex1(A) ) ), file(prepower,k1_prepower), []). fof(redefinition_k1_prob_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A)) & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) & m1_subset_1(C,k5_numbers) ) => k1_prob_1(A,B,C) = k1_funct_1(B,C) ) ), file(prob_1,k1_prob_1), []). fof(redefinition_k1_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k1_real_1(A) = k4_xcmplx_0(A) ) ), file(real_1,k1_real_1), []). fof(redefinition_k1_seqfunc,axiom,( ! [A,B,C,D] : ( ( m1_seqfunc(C,A,B) & v4_ordinal2(D) ) => k1_seqfunc(A,B,C,D) = k1_funct_1(C,D) ) ), file(seqfunc,k1_seqfunc), []). fof(redefinition_k2_binop_1,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_subset_1(E,A) & m1_subset_1(F,B) ) => k2_binop_1(A,B,C,D,E,F) = k1_binop_1(D,E,F) ) ), file(binop_1,k2_binop_1), []). fof(redefinition_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), []). fof(redefinition_k2_nagata_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m1_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => k2_nagata_1(A,B) = k3_card_3(B) ) ), file(nagata_1,k2_nagata_1), []). fof(redefinition_k2_prob_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A)) & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) ) => k2_prob_1(A,B) = k3_card_3(B) ) ), file(prob_1,k2_prob_1), []). 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_cantor_1,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) & v1_funct_1(D) & v1_funct_2(D,B,k1_zfmisc_1(C)) & m1_relset_1(D,B,k1_zfmisc_1(C)) ) => k3_cantor_1(A,B,C,D,E) = k1_funct_1(D,E) ) ), file(cantor_1,k3_cantor_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_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), []). fof(redefinition_k3_square_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_square_1(A,B) = k1_square_1(A,B) ) ), file(square_1,k3_square_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_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_finsop_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k5_finsop_1(A) = k1_relat_1(A) ) ), file(finsop_1,k5_finsop_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), []). fof(redefinition_k5_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_urysohn3,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v2_pre_topc(B) & l1_pre_topc(B) & v1_funct_1(C) & v1_funct_2(C,A,k1_pcomps_1(u1_struct_0(B))) & m1_relset_1(C,A,k1_pcomps_1(u1_struct_0(B))) & m1_subset_1(D,A) ) => k5_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) ), file(urysohn3,k5_urysohn3), []). fof(redefinition_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), []). fof(redefinition_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_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_k7_urysohn3,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) & 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(D,u1_struct_0(A)) ) => k7_urysohn3(A,B,C,D) = k1_funct_1(C,D) ) ), file(urysohn3,k7_urysohn3), []). 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_k9_urysohn1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_subset_1(C,u1_struct_0(A)) ) => k9_urysohn1(A,B,C) = k1_funct_1(B,C) ) ), file(urysohn1,k9_urysohn1), []). fof(redefinition_k9_urysohn3,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m1_subset_1(C,u1_struct_0(A)) ) => k9_urysohn3(A,B,C) = k1_funct_1(B,C) ) ), file(urysohn3,k9_urysohn3), []). 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_fraenkel,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(B) & m1_fraenkel(C,A,B) ) => ! [D] : ( m2_fraenkel(D,A,B,C) <=> m1_subset_1(D,C) ) ) ), file(fraenkel,m2_fraenkel), []). 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_binop_1__e22_23_1_1__nagata_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( ! [B] : ( m1_subset_1(B,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) => ! [C] : ( m1_subset_1(C,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) => ? [D] : ( m1_subset_1(D,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & k3_nagata_1(k6_borsuk_1(A,A),B,C) = D ) ) ) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ! [C] : ( m1_subset_1(C,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) => ! [D] : ( m1_subset_1(D,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) => k3_nagata_1(k6_borsuk_1(A,A),C,D) = k2_binop_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),B,C,D) ) ) ) ) ) ), file(nagata_2,s1_binop_1__e22_23_1_1__nagata_2), []). fof(s1_funct_2__e26_23_1_1__nagata_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k6_borsuk_1(A,A)),k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers))) & m2_relset_1(B,u1_struct_0(k6_borsuk_1(A,A)),k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers))) & v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(C,k2_zfmisc_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ! [D] : ~ ( r2_hidden(D,u1_struct_0(k6_borsuk_1(A,A))) & ! [E] : ~ ( r2_hidden(E,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ! [F] : ( m2_finseq_1(F,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) => ( F = k1_funct_1(B,D) => E = k2_finsop_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),F,C) ) ) ) ) => ? [D] : ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(k6_borsuk_1(A,A)),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(D,u1_struct_0(k6_borsuk_1(A,A)),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ! [E] : ( r2_hidden(E,u1_struct_0(k6_borsuk_1(A,A))) => ! [G] : ( m2_finseq_1(G,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) => ( G = k1_funct_1(B,E) => k1_funct_1(D,E) = k2_finsop_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),G,C) ) ) ) ) ) ) ), file(nagata_2,s1_funct_2__e26_23_1_1__nagata_2), []). fof(s1_funct_2__e32_23_1_1__nagata_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(B,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) ) => ( ! [C] : ~ ( r2_hidden(C,u1_struct_0(k6_borsuk_1(A,A))) & ! [D] : ~ ( r2_hidden(D,k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,A)))) & ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ( C = k1_domain_1(u1_struct_0(A),u1_struct_0(A),E,F) => D = k7_borsuk_1(A,A,k8_funct_2(u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)),B,E),k8_funct_2(u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)),B,F)) ) ) ) ) ) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,A)),k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,A)))) & m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,A)),k1_pcomps_1(u1_struct_0(k6_borsuk_1(A,A)))) & ! [D] : ( r2_hidden(D,u1_struct_0(k6_borsuk_1(A,A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => ! [H] : ( m1_subset_1(H,u1_struct_0(A)) => ( D = k1_domain_1(u1_struct_0(A),u1_struct_0(A),G,H) => k1_funct_1(C,D) = k7_borsuk_1(A,A,k8_funct_2(u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)),B,G),k8_funct_2(u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)),B,H)) ) ) ) ) ) ) ) ), file(nagata_2,s1_funct_2__e32_23_1_1__nagata_2), []). fof(s1_funct_2__e7_23_1_1__nagata_2,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,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_funct_1(D) & v1_funct_2(D,k1_pcomps_1(u1_struct_0(A)),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(D,k1_pcomps_1(u1_struct_0(A)),k1_pcomps_1(u1_struct_0(A))) ) => ( ! [E] : ~ ( r2_hidden(E,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C)) & ! [F] : ~ ( r2_hidden(F,k1_fraenkel(u1_struct_0(A),k1_numbers)) & ? [G] : ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(A),k1_numbers) & m2_relset_1(G,u1_struct_0(A),k1_numbers) & ? [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(A))) & H = E & G = F & v9_pscomp_1(G,A) & ! [I] : ( m1_subset_1(I,u1_struct_0(A)) => ( r1_xreal_0(0,k8_funct_2(u1_struct_0(A),k1_numbers,G,I)) & r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,G,I),1) & ( r2_hidden(I,k3_subset_1(u1_struct_0(A),H)) => k8_funct_2(u1_struct_0(A),k1_numbers,G,I) = 0 ) & ( r2_hidden(I,k6_pre_topc(A,k8_funct_2(k1_zfmisc_1(u1_struct_0(A)),k1_pcomps_1(u1_struct_0(A)),D,H))) => k8_funct_2(u1_struct_0(A),k1_numbers,G,I) = 1 ) ) ) ) ) ) ) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C),k1_fraenkel(u1_struct_0(A),k1_numbers)) & m2_relset_1(E,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C),k1_fraenkel(u1_struct_0(A),k1_numbers)) & ! [F] : ( r2_hidden(F,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C)) => ? [J] : ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(A),k1_numbers) & m2_relset_1(J,u1_struct_0(A),k1_numbers) & ? [K] : ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A))) & K = F & J = k1_funct_1(E,F) & v9_pscomp_1(J,A) & ! [L] : ( m1_subset_1(L,u1_struct_0(A)) => ( r1_xreal_0(0,k8_funct_2(u1_struct_0(A),k1_numbers,J,L)) & r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,J,L),1) & ( r2_hidden(L,k3_subset_1(u1_struct_0(A),K)) => k8_funct_2(u1_struct_0(A),k1_numbers,J,L) = 0 ) & ( r2_hidden(L,k6_pre_topc(A,k8_funct_2(k1_zfmisc_1(u1_struct_0(A)),k1_pcomps_1(u1_struct_0(A)),D,K))) => k8_funct_2(u1_struct_0(A),k1_numbers,J,L) = 1 ) ) ) ) ) ) ) ) ) ), file(nagata_2,s1_funct_2__e7_23_1_1__nagata_2), []). fof(s1_funct_2__e9_23_1__nagata_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => ( ! [C] : ~ ( r2_hidden(C,k5_numbers) & ! [D] : ~ ( r2_hidden(D,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ? [E] : ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & E = D & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( k1_funct_1(k2_funct_1(k1_nagata_2),C) = k1_domain_1(k5_numbers,k5_numbers,F,G) => ? [H] : ( v1_funct_1(H) & v1_funct_2(H,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(H,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & E = H & v9_pscomp_1(E,k6_borsuk_1(A,A)) & r1_nagata_1(u1_struct_0(A),H) & ! [I] : ( m1_subset_1(I,u1_struct_0(A)) => ! [J] : ( m1_subset_1(J,u1_struct_0(A)) => ( r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,H,k1_domain_1(u1_struct_0(A),u1_struct_0(A),I,J)),1) & ! [K] : ( m1_subset_1(K,u1_struct_0(A)) => ! [L] : ( m1_subset_1(L,u1_struct_0(A)) => ( ? [M] : ( m1_subset_1(M,k1_zfmisc_1(u1_struct_0(A))) & ? [N] : ( m1_subset_1(N,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(M,A) & v3_pre_topc(N,A) & r2_hidden(M,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,G)) & r2_hidden(N,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,F)) & r2_hidden(K,M) & r1_tarski(k6_pre_topc(A,M),N) & ~ r2_hidden(L,N) ) ) => k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,H,k1_domain_1(u1_struct_0(A),u1_struct_0(A),K,L)) = 1 ) ) ) ) ) ) ) ) ) ) ) ) ) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(C,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ! [D] : ( r2_hidden(D,k5_numbers) => ? [O] : ( v1_funct_1(O) & v1_funct_2(O,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(O,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & O = k1_funct_1(C,D) & ! [P] : ( m2_subset_1(P,k1_numbers,k5_numbers) => ! [Q] : ( m2_subset_1(Q,k1_numbers,k5_numbers) => ( k1_funct_1(k2_funct_1(k1_nagata_2),D) = k1_domain_1(k5_numbers,k5_numbers,P,Q) => ? [R] : ( v1_funct_1(R) & v1_funct_2(R,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(R,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & O = R & v9_pscomp_1(O,k6_borsuk_1(A,A)) & r1_nagata_1(u1_struct_0(A),R) & ! [S] : ( m1_subset_1(S,u1_struct_0(A)) => ! [T] : ( m1_subset_1(T,u1_struct_0(A)) => ( r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,R,k1_domain_1(u1_struct_0(A),u1_struct_0(A),S,T)),1) & ! [U] : ( m1_subset_1(U,u1_struct_0(A)) => ! [V] : ( m1_subset_1(V,u1_struct_0(A)) => ( ? [W] : ( m1_subset_1(W,k1_zfmisc_1(u1_struct_0(A))) & ? [X] : ( m1_subset_1(X,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(W,A) & v3_pre_topc(X,A) & r2_hidden(W,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,Q)) & r2_hidden(X,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,P)) & r2_hidden(U,W) & r1_tarski(k6_pre_topc(A,W),X) & ~ r2_hidden(V,X) ) ) => k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,R,k1_domain_1(u1_struct_0(A),u1_struct_0(A),U,V)) = 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), file(nagata_2,s1_funct_2__e9_23_1__nagata_2), []). fof(s1_nat_1__e31_23_1_1_19__nagata_2,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(k6_borsuk_1(A,A))) & m2_finseq_1(C,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ( ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(E,0) & r1_xreal_0(0,k3_finseq_1(C)) ) => ( E = 0 | ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(G,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( ( F = k1_funct_1(C,E) & G = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,0) ) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,F,B),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,G,B)) ) ) ) ) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ( ( r1_xreal_0(I,H) & r1_xreal_0(H,k3_finseq_1(C)) ) => ( I = 0 | ! [J] : ( ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(J,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ! [K] : ( ( v1_funct_1(K) & v1_funct_2(K,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(K,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( ( J = k1_funct_1(C,I) & K = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,H) ) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,J,B),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,K,B)) ) ) ) ) ) ) => ! [L] : ( m2_subset_1(L,k1_numbers,k5_numbers) => ( ( r1_xreal_0(L,k1_nat_1(H,1)) & r1_xreal_0(k1_nat_1(H,1),k3_finseq_1(C)) ) => ( L = 0 | ! [M] : ( ( v1_funct_1(M) & v1_funct_2(M,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(M,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ! [N] : ( ( v1_funct_1(N) & v1_funct_2(N,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(N,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( ( M = k1_funct_1(C,L) & N = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,k1_nat_1(H,1)) ) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,M,B),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,N,B)) ) ) ) ) ) ) ) ) ) => ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ! [O] : ( m2_subset_1(O,k1_numbers,k5_numbers) => ( ( r1_xreal_0(O,H) & r1_xreal_0(H,k3_finseq_1(C)) ) => ( O = 0 | ! [P] : ( ( v1_funct_1(P) & v1_funct_2(P,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(P,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ! [Q] : ( ( v1_funct_1(Q) & v1_funct_2(Q,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(Q,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( ( P = k1_funct_1(C,O) & Q = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,H) ) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,P,B),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,Q,B)) ) ) ) ) ) ) ) ) ) ), file(nagata_2,s1_nat_1__e31_23_1_1_19__nagata_2), []). fof(s1_nat_1__e7_23_1_1_17_2_1_1_2__nagata_2,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m2_finseq_1(B,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m1_subset_1(C,u1_struct_0(k6_borsuk_1(A,A))) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ( ( r1_xreal_0(0,k3_finseq_1(B)) => ( 0 = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,0),C) = 0 ) ) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(E,k3_finseq_1(B)) => ( E = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,E),C) = 0 ) ) => ( r1_xreal_0(k1_nat_1(E,1),k3_finseq_1(B)) => ( k1_nat_1(E,1) = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,k1_nat_1(E,1)),C) = 0 ) ) ) ) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_xreal_0(E,k3_finseq_1(B)) => ( E = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,E),C) = 0 ) ) ) ) ) ), file(nagata_2,s1_nat_1__e7_23_1_1_17_2_1_1_2__nagata_2), []). fof(s1_nat_1__e7_23_1_1_17_4_1_2__nagata_2,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(k6_borsuk_1(A,A))) & m2_finseq_1(C,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ( ( r1_xreal_0(0,k3_finseq_1(C)) => ( 0 = 0 | ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(E,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( E = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,0) => r1_xreal_0(0,k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,E,B)) ) ) ) ) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( ( r1_xreal_0(F,k3_finseq_1(C)) => ( F = 0 | ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(G,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( G = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,F) => r1_xreal_0(0,k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,G,B)) ) ) ) ) => ( r1_xreal_0(k1_nat_1(F,1),k3_finseq_1(C)) => ( k1_nat_1(F,1) = 0 | ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(H,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( H = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,k1_nat_1(F,1)) => r1_xreal_0(0,k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,H,B)) ) ) ) ) ) ) ) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( r1_xreal_0(F,k3_finseq_1(C)) => ( F = 0 | ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(I,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( I = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,F) => r1_xreal_0(0,k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,I,B)) ) ) ) ) ) ) ) ), file(nagata_2,s1_nat_1__e7_23_1_1_17_4_1_2__nagata_2), []). fof(s1_nat_1__e7_23_1_1_18_1_1_1_2__nagata_2,axiom,( ! [A,B,C,D] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(k6_borsuk_1(A,A))) & m2_finseq_1(C,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(D,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ( ( r1_xreal_0(0,k3_finseq_1(C)) => ( 0 = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,0),B) = 0 ) ) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(E,k3_finseq_1(C)) => ( E = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,E),B) = 0 ) ) => ( r1_xreal_0(k1_nat_1(E,1),k3_finseq_1(C)) => ( k1_nat_1(E,1) = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,k1_nat_1(E,1)),B) = 0 ) ) ) ) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r1_xreal_0(E,k3_finseq_1(C)) => ( E = 0 | k1_funct_1(k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),D,E),B) = 0 ) ) ) ) ) ), file(nagata_2,s1_nat_1__e7_23_1_1_18_1_1_1_2__nagata_2), []). fof(s1_nat_1__e7_23_1_1_18_2_1_1_2__nagata_2,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(k6_borsuk_1(A,A))) & m1_subset_1(C,u1_struct_0(k6_borsuk_1(A,A))) & m1_subset_1(D,u1_struct_0(k6_borsuk_1(A,A))) & m2_finseq_1(E,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(F,k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ( ( r1_xreal_0(0,k3_finseq_1(E)) => ( 0 = 0 | ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(G,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( G = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),F,0) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,G,B),k3_real_1(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,G,C),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,G,D))) ) ) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( ( r1_xreal_0(H,k3_finseq_1(E)) => ( H = 0 | ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(I,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( I = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),F,H) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,I,B),k3_real_1(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,I,C),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,I,D))) ) ) ) ) => ( r1_xreal_0(k1_nat_1(H,1),k3_finseq_1(E)) => ( k1_nat_1(H,1) = 0 | ! [J] : ( ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(J,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( J = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),F,k1_nat_1(H,1)) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,J,B),k3_real_1(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,J,C),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,J,D))) ) ) ) ) ) ) ) => ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( r1_xreal_0(H,k3_finseq_1(E)) => ( H = 0 | ! [K] : ( ( v1_funct_1(K) & v1_funct_2(K,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(K,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( K = k8_funct_2(k5_numbers,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),F,H) => r1_xreal_0(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,K,B),k3_real_1(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,K,C),k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,K,D))) ) ) ) ) ) ) ) ), file(nagata_2,s1_nat_1__e7_23_1_1_18_2_1_1_2__nagata_2), []). fof(s21_fraenkel__e2_23_1_1_4__nagata_2,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,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m1_subset_1(E,u1_struct_0(A)) ) => ( v1_finset_1(a_5_0_nagata_2(A,B,C,D,E)) => v1_finset_1(a_0_42_fraenkel) ) ) ), file(nagata_2,s21_fraenkel__e2_23_1_1_4__nagata_2), []). fof(s21_fraenkel__e4_23_1_1_7__nagata_2,axiom,( ! [A,B,C,D,E,F,G] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_funct_1(D) & v1_funct_2(D,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C),k1_fraenkel(u1_struct_0(A),k1_numbers)) & m2_relset_1(D,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C),k1_fraenkel(u1_struct_0(A),k1_numbers)) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(E,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m1_subset_1(F,u1_struct_0(A)) & m1_subset_1(G,u1_struct_0(A)) ) => ( v1_finset_1(k2_xboole_0(a_6_7_nagata_2(A,B,C,D,E,F),a_6_7_nagata_2(A,B,C,D,E,G))) => v1_finset_1(a_0_42_fraenkel) ) ) ), file(nagata_2,s21_fraenkel__e4_23_1_1_7__nagata_2), []). fof(s2_funct_2__e3_23_1_1__nagata_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( ! [D] : ( r2_hidden(D,k1_pcomps_1(u1_struct_0(A))) => r2_hidden(k3_tarski(a_4_0_nagata_2(A,B,C,D)),k1_pcomps_1(u1_struct_0(A))) ) => ? [D] : ( v1_funct_1(D) & v1_funct_2(D,k1_pcomps_1(u1_struct_0(A)),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(D,k1_pcomps_1(u1_struct_0(A)),k1_pcomps_1(u1_struct_0(A))) & ! [E] : ( r2_hidden(E,k1_pcomps_1(u1_struct_0(A))) => k1_funct_1(D,E) = k3_tarski(a_4_0_nagata_2(A,B,C,E)) ) ) ) ) ), file(nagata_2,s2_funct_2__e3_23_1_1__nagata_2), []). fof(s3_funct_2__e10_23_1_1__nagata_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ? [E] : ( m1_subset_1(E,k1_pcomps_1(u1_struct_0(A))) & r2_hidden(D,E) & v3_pre_topc(E,A) & v1_finset_1(a_4_6_nagata_2(A,B,C,E)) ) ) => ? [D] : ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( r2_hidden(E,k8_funct_2(u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)),D,E)) & v3_pre_topc(k8_funct_2(u1_struct_0(A),k1_pcomps_1(u1_struct_0(A)),D,E),A) & v1_finset_1(a_5_0_nagata_2(A,B,C,D,E)) ) ) ) ) ) ), file(nagata_2,s3_funct_2__e10_23_1_1__nagata_2), []). fof(s3_funct_2__e14_23_1_1__nagata_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( ! [B] : ( m1_subset_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers)) => ? [C] : ( m1_subset_1(C,k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => k1_funct_1(C,k1_domain_1(u1_struct_0(A),u1_struct_0(A),D,E)) = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,D),k8_funct_2(u1_struct_0(A),k1_numbers,B,E))) ) ) ) ) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & ! [C] : ( m1_subset_1(C,k1_fraenkel(u1_struct_0(A),k1_numbers)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => k1_funct_1(k8_funct_2(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers),B,C),k1_domain_1(u1_struct_0(A),u1_struct_0(A),F,G)) = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,C,F),k8_funct_2(u1_struct_0(A),k1_numbers,C,G))) ) ) ) ) ) ) ), file(nagata_2,s3_funct_2__e14_23_1_1__nagata_2), []). fof(s3_funct_2__e19_23_1_1__nagata_2,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_subset_1(C,k1_numbers,k5_numbers) & v1_funct_1(D) & v1_funct_2(D,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C),k1_fraenkel(u1_struct_0(A),k1_numbers)) & m2_relset_1(D,k3_cantor_1(u1_struct_0(A),k5_numbers,k1_pcomps_1(u1_struct_0(A)),B,C),k1_fraenkel(u1_struct_0(A),k1_numbers)) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(E,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & v1_funct_1(F) & v1_funct_2(F,k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) & m2_relset_1(F,k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)) ) => ( ! [G] : ( m1_subset_1(G,u1_struct_0(k6_borsuk_1(A,A))) => ? [H] : ( m1_subset_1(H,k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers))) & v2_funct_1(H) & ? [I] : ( m1_subset_1(I,u1_struct_0(A)) & ? [J] : ( m1_subset_1(J,u1_struct_0(A)) & k1_domain_1(u1_struct_0(A),u1_struct_0(A),I,J) = G & k2_relat_1(H) = a_8_0_nagata_2(A,B,C,D,E,F,I,J) ) ) ) ) => ? [G] : ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k6_borsuk_1(A,A)),k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers))) & m2_relset_1(G,u1_struct_0(k6_borsuk_1(A,A)),k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers))) & ! [H] : ( m1_subset_1(H,u1_struct_0(k6_borsuk_1(A,A))) => ( v2_funct_1(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)),G,H)) & ? [K] : ( m1_subset_1(K,u1_struct_0(A)) & ? [L] : ( m1_subset_1(L,u1_struct_0(A)) & k1_domain_1(u1_struct_0(A),u1_struct_0(A),K,L) = H & k2_relat_1(k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k13_finseq_1(k1_fraenkel(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers)),G,H)) = a_8_0_nagata_2(A,B,C,D,E,F,K,L) ) ) ) ) ) ) ) ), file(nagata_2,s3_funct_2__e19_23_1_1__nagata_2), []). fof(s7_funct_2__e2_23_1_1_5__nagata_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m2_fraenkel(B,u1_struct_0(A),k1_numbers,k1_fraenkel(u1_struct_0(A),k1_numbers)) ) => ( ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ? [E] : ( m1_subset_1(E,k1_numbers) & E = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,C),k8_funct_2(u1_struct_0(A),k1_numbers,B,D))) ) ) ) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => k1_funct_1(C,k4_tarski(D,E)) = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,D),k8_funct_2(u1_struct_0(A),k1_numbers,B,E))) ) ) ) ) ) ), file(nagata_2,s7_funct_2__e2_23_1_1_5__nagata_2), []). 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(symmetry_r1_xboole_0,axiom,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), []). 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(t10_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(C,D) & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ), file(xreal_1,t10_xreal_1), []). fof(t11_binop_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m2_relset_1(B,k2_zfmisc_1(A,A),A) ) => ! [C] : ( m1_subset_1(C,A) => ( r3_binop_1(A,C,B) <=> ! [D] : ( m1_subset_1(D,A) => ( k1_binop_1(B,C,D) = D & k1_binop_1(B,D,C) = D ) ) ) ) ) ), file(binop_1,t11_binop_1), []). fof(t11_funct_2,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => r2_hidden(C,k1_funct_2(A,B)) ) ) ), file(funct_2,t11_funct_2), []). fof(t121_funct_2,axiom,( ! [A,B,C] : ( r2_hidden(C,k1_funct_2(A,B)) => ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) ) ), file(funct_2,t121_funct_2), []). fof(t12_pcomps_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( ( r1_tarski(B,C) & v1_pcomps_1(C,A) ) => v1_pcomps_1(B,A) ) ) ) ) ), file(pcomps_1,t12_pcomps_1), []). fof(t132_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => r1_xreal_0(0,k17_complex1(A)) ) ), file(complex1,t132_complex1), []). fof(t138_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k4_xcmplx_0(A)) = k17_complex1(A) ) ), file(complex1,t138_complex1), []). fof(t13_finset_1,axiom,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), []). fof(t142_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => r1_xreal_0(k17_complex1(k2_xcmplx_0(A,B)),k3_real_1(k17_complex1(A),k17_complex1(B))) ) ) ), file(complex1,t142_complex1), []). fof(t14_finset_1,axiom,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,t14_finset_1), []). fof(t15_finset_1,axiom,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,t15_finset_1), []). fof(t15_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v3_pcomps_1(A) => ( v4_compts_1(A) & v1_urysohn1(A) ) ) ) ), file(nagata_1,t15_nagata_1), []). fof(t16_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ~ ( v3_pcomps_1(A) & ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => ~ v6_nagata_1(B,A) ) ) ) ), file(nagata_1,t16_nagata_1), []). fof(t17_nagata_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_urysohn1(A) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_seqfunc(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) => ( ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ? [E] : ( v1_funct_1(E) & v1_funct_2(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = E & r1_nagata_1(u1_struct_0(A),E) & ! [F] : ( m1_subset_1(F,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) => r1_xreal_0(k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,E,F),B) ) & ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(F,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( E = F => v9_pscomp_1(F,k6_borsuk_1(A,A)) ) ) ) ) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( ( ~ v1_xboole_0(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( ~ r2_hidden(D,E) & v4_pre_topc(E,A) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ? [G] : ( v1_funct_1(G) & v1_funct_2(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(G,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,F) = G & r1_xreal_0(k8_funct_2(u1_struct_0(A),k1_numbers,k3_nagata_2(u1_struct_0(A),G,E),D),0) ) ) ) ) ) ) => ( ? [D] : ( v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & r1_pcomps_1(u1_struct_0(A),D) & ! [E] : ( m1_subset_1(E,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) => k8_funct_2(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,D,E) = k2_series_1(k11_seq_1(k2_prepower(k6_real_1(1,2)),k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),C,E))) ) ) & v3_pcomps_1(A) ) ) ) ) ) ) ), file(nagata_2,t17_nagata_2), []). fof(t17_xboole_1,axiom,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_xboole_1), []). fof(t18_nagata_2,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_finseq_1(C,A) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,A),A) & m2_relset_1(D,k2_zfmisc_1(A,A),A) ) => ~ ( v2_funct_1(B) & v2_funct_1(C) & r1_tarski(k5_relset_1(k5_numbers,A,C),k5_relset_1(k5_numbers,A,B)) & v1_binop_1(D,A) & v2_binop_1(D,A) & ( v1_setwiseo(D,A) | ( r1_xreal_0(1,k3_finseq_1(C)) & ~ r1_xreal_0(k3_finseq_1(B),k3_finseq_1(C)) ) ) & ! [E] : ( m2_finseq_1(E,A) => ~ ( v2_funct_1(E) & k5_relset_1(k5_numbers,A,E) = k6_subset_1(A,k5_relset_1(k5_numbers,A,B),k5_relset_1(k5_numbers,A,C)) & k2_finsop_1(A,B,D) = k2_binop_1(A,A,A,D,k2_finsop_1(A,C,D),k2_finsop_1(A,E,D)) ) ) ) ) ) ) ) ), file(nagata_2,t18_nagata_2), []). fof(t19_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v4_compts_1(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => ( m1_cantor_1(k2_nagata_1(A,B),A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ~ ( v3_pre_topc(C,A) & r2_hidden(D,C) & ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r2_hidden(D,E) & r1_tarski(k6_pre_topc(A,E),C) & r2_hidden(E,k2_nagata_1(A,B)) ) ) ) ) ) ) ) ) ) ), file(nagata_1,t19_nagata_1), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_boole,axiom,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_seqfunc,axiom,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( m1_seqfunc(C,A,B) <=> ( k1_relat_1(C) = k5_numbers & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( v1_funct_1(k1_funct_1(C,D)) & m2_relset_1(k1_funct_1(C,D),A,B) ) ) ) ) ) ), file(seqfunc,t1_seqfunc), []). 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_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( ( v4_compts_1(A) & v1_urysohn1(A) ) => ( ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(u1_struct_0(A)))) ) => ~ v6_nagata_1(B,A) ) | v5_compts_1(A) ) ) ) ), file(nagata_1,t20_nagata_1), []). fof(t21_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m2_relset_1(B,u1_struct_0(A),k1_numbers) ) => ( v9_pscomp_1(B,A) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) & m2_relset_1(C,u1_struct_0(k6_borsuk_1(A,A)),k1_numbers) ) => ( ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => k8_funct_2(u1_struct_0(k6_borsuk_1(A,A)),k1_numbers,C,k8_borsuk_1(A,A,D,E)) = k18_complex1(k5_real_1(k8_funct_2(u1_struct_0(A),k1_numbers,B,D),k8_funct_2(u1_struct_0(A),k1_numbers,B,E))) ) ) => v9_pscomp_1(C,k6_borsuk_1(A,A)) ) ) ) ) ) ), file(nagata_1,t21_nagata_1), []). fof(t23_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) ) => ( ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m2_relset_1(C,u1_struct_0(A),k1_numbers) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_numbers) & m2_relset_1(D,u1_struct_0(A),k1_numbers) ) => k1_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) ) => ( v1_setwiseo(B,k1_fraenkel(u1_struct_0(A),k1_numbers)) & v1_binop_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers)) & v2_binop_1(B,k1_fraenkel(u1_struct_0(A),k1_numbers)) ) ) ) ) ), file(nagata_1,t23_nagata_1), []). fof(t23_pcomps_1,axiom,( ! [A] : ( ( ~ v3_struct_0(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_pcomps_1(B,A) => k6_pre_topc(A,k5_setfam_1(u1_struct_0(A),B)) = k5_setfam_1(u1_struct_0(A),k3_pcomps_1(A,B)) ) ) ) ), file(pcomps_1,t23_pcomps_1), []). fof(t23_urysohn3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v5_compts_1(A) & l1_pre_topc(A) ) => ! [B] : ( ( v4_pre_topc(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( ( v4_pre_topc(C,A) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) => ~ ( r1_xboole_0(B,C) & ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_topmetr)) ) => ~ ( v5_pre_topc(D,A,k3_topmetr) & ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( r1_xreal_0(0,k9_urysohn3(A,D,E)) & r1_xreal_0(k9_urysohn3(A,D,E),1) & ( r2_hidden(E,B) => k9_urysohn3(A,D,E) = 0 ) & ( r2_hidden(E,C) => k9_urysohn3(A,D,E) = 1 ) ) ) ) ) ) ) ) ) ), file(urysohn3,t23_urysohn3), []). fof(t24_topmetr,axiom,( u1_struct_0(k3_topmetr) = k1_numbers ), file(topmetr,t24_topmetr), []). fof(t25_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( k3_finseq_1(A) = 0 <=> A = k1_xboole_0 ) ) ), file(finseq_1,t25_finseq_1), []). fof(t25_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) & m2_relset_1(B,k2_zfmisc_1(k1_fraenkel(u1_struct_0(A),k1_numbers),k1_fraenkel(u1_struct_0(A),k1_numbers)),k1_fraenkel(u1_struct_0(A),k1_numbers)) ) => ( ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m2_relset_1(C,u1_struct_0(A),k1_numbers) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_numbers) & m2_relset_1(D,u1_struct_0(A),k1_numbers) ) => k1_binop_1(B,C,D) = k3_nagata_1(A,C,D) ) ) => ! [C] : ( m2_finseq_1(C,k1_fraenkel(u1_struct_0(A),k1_numbers)) => ( ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(D,k3_finseq_1(C)) => ( 0 = D | ( v1_funct_1(k1_funct_1(C,D)) & v1_funct_2(k1_funct_1(C,D),u1_struct_0(A),k1_numbers) & v9_pscomp_1(k1_funct_1(C,D),A) & m2_relset_1(k1_funct_1(C,D),u1_struct_0(A),k1_numbers) ) ) ) ) => ( v1_funct_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B)) & v1_funct_2(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),u1_struct_0(A),k1_numbers) & v9_pscomp_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),A) & m2_relset_1(k2_finsop_1(k1_fraenkel(u1_struct_0(A),k1_numbers),C,B),u1_struct_0(A),k1_numbers) ) ) ) ) ) ) ), file(nagata_1,t25_nagata_1), []). fof(t25_prob_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_zfmisc_1(A)) & m2_relset_1(C,k5_numbers,k1_zfmisc_1(A)) ) => ( r2_hidden(B,k2_prob_1(A,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & r2_hidden(B,k1_prob_1(A,C,D)) ) ) ) ), file(prob_1,t25_prob_1), []). fof(t26_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) & m2_relset_1(C,u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B))) ) => ( ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( v1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D)) & v1_funct_2(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),A,B) & m2_relset_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,D),u1_struct_0(A),u1_struct_0(B)) ) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) & m2_relset_1(D,u1_struct_0(A),k1_pcomps_1(u1_struct_0(A))) ) => ( ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( r2_hidden(E,k5_urysohn3(u1_struct_0(A),A,D,E)) & v3_pre_topc(k5_urysohn3(u1_struct_0(A),A,D,E),A) & ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => ! [G] : ( m1_subset_1(G,u1_struct_0(A)) => ( r2_hidden(F,k5_urysohn3(u1_struct_0(A),A,D,G)) => k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,F),F) = k1_funct_1(k8_funct_2(u1_struct_0(A),k1_fraenkel(u1_struct_0(A),u1_struct_0(B)),C,G),F) ) ) ) ) ) => v5_pre_topc(k4_nagata_1(A,B,D,C),A,B) ) ) ) ) ) ) ), file(nagata_1,t26_nagata_1), []). 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(t27_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( A = k1_xboole_0 <=> k2_relat_1(A) = k1_xboole_0 ) ) ), file(finseq_1,t27_finseq_1), []). 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_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),k1_numbers) & m2_relset_1(C,u1_struct_0(A),k1_numbers) ) => ( v9_pscomp_1(C,A) => v9_pscomp_1(k5_nagata_1(u1_struct_0(A),B,C),A) ) ) ) ) ), file(nagata_1,t27_nagata_1), []). fof(t28_nagata_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( r1_nagata_1(A,B) <=> ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => ! [E] : ( m1_subset_1(E,A) => ( k1_metric_1(A,A,B,C,C) = 0 & r1_xreal_0(k1_metric_1(A,A,B,C,E),k3_real_1(k1_metric_1(A,A,B,C,D),k1_metric_1(A,A,B,E,D))) ) ) ) ) ) ) ), file(nagata_1,t28_nagata_1), []). fof(t28_xboole_1,axiom,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), []). fof(t29_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v4_pre_topc(B,A) <=> v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_1,t29_tops_1), []). fof(t2_boole,axiom,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), []). fof(t2_nagata_2,axiom,( v3_funct_2(k1_nagata_2,k2_zfmisc_1(k5_numbers,k5_numbers),k5_numbers) ), file(nagata_2,t2_nagata_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(t2_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(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), []). fof(t30_nagata_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) => ( r1_nagata_1(u1_struct_0(A),C) => ( r1_xreal_0(B,0) | r1_nagata_1(u1_struct_0(A),k5_nagata_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) ) ) ) ) ), file(nagata_1,t30_nagata_1), []). fof(t33_zfmisc_1,axiom,( ! [A,B,C,D] : ( k4_tarski(A,B) = k4_tarski(C,D) => ( A = C & B = D ) ) ), file(zfmisc_1,t33_zfmisc_1), []). fof(t35_square_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => r1_xreal_0(k1_square_1(A,B),A) ) ) ), file(square_1,t35_square_1), []). fof(t37_xboole_1,axiom,( ! [A,B] : ( k4_xboole_0(A,B) = k1_xboole_0 <=> r1_tarski(A,B) ) ), file(xboole_1,t37_xboole_1), []). 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(t39_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( ~ r1_xreal_0(1,A) => A = 0 ) ) ), file(nat_1,t39_nat_1), []). fof(t3_boole,axiom,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), []). fof(t3_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_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_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(t43_subset_1,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => ( r1_xboole_0(B,C) <=> r1_tarski(B,k3_subset_1(A,C)) ) ) ) ), file(subset_1,t43_subset_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(t47_xboole_1,axiom,( ! [A,B] : k4_xboole_0(A,k3_xboole_0(A,B)) = k4_xboole_0(A,B) ), file(xboole_1,t47_xboole_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_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). 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(t54_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( B = k2_funct_1(A) <=> ( k1_relat_1(B) = k2_relat_1(A) & ! [C,D] : ( ( ( r2_hidden(C,k2_relat_1(A)) & D = k1_funct_1(B,C) ) => ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) & ( ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) => ( r2_hidden(C,k2_relat_1(A)) & D = k1_funct_1(B,C) ) ) ) ) ) ) ) ) ), file(funct_1,t54_funct_1), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t60_seq_4,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_membered(A) ) => ! [B] : ( v1_xreal_0(B) => ( ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(B,C) ) ) => r1_xreal_0(B,k3_seq_4(A)) ) ) ) ), file(seq_4,t60_seq_4), []). fof(t63_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_xboole_0(B,C) ) => r1_xboole_0(A,C) ) ), file(xboole_1,t63_xboole_1), []). fof(t6_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(t73_finseq_4,axiom,( ! [A] : ~ ( v1_finset_1(A) & ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ~ ( k2_relat_1(B) = A & v2_funct_1(B) ) ) ) ), file(finseq_4,t73_finseq_4), []). fof(t7_absvalue,axiom,( ! [A] : ( v1_xreal_0(A) => ( A = 0 <=> k18_complex1(A) = 0 ) ) ), file(absvalue,t7_absvalue), []). 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(t83_topreal6,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(A),k1_numbers) & m2_relset_1(B,u1_struct_0(A),k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(k3_topmetr)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(k3_topmetr)) ) => ( B = C => ( v9_pscomp_1(B,A) <=> v5_pre_topc(C,A,k3_topmetr) ) ) ) ) ) ), file(topreal6,t83_topreal6), []). 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(t92_zfmisc_1,axiom,( ! [A,B] : ( r2_hidden(A,B) => r1_tarski(A,k3_tarski(B)) ) ), file(zfmisc_1,t92_zfmisc_1), []). fof(t9_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), []).