% Mizar problem: t15_nagata_2,nagata_2,1080,57 fof(t15_nagata_2,conjecture,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(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] : ( ( 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) ) => ( ! [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))) ) => ( r1_nagata_1(u1_struct_0(A),D) & ! [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) ) => ( D = E => v9_pscomp_1(E,k6_borsuk_1(A,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_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_k2_tarski,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,commutativity_k5_seq_1,commutativity_k8_seq_1,connectedness_r1_xreal_0,d11_seqfunc,d1_absvalue,d1_binop_1,d1_funct_2,d1_prepower,d2_zfmisc_1,d5_borsuk_1,d5_tarski,d6_seq_1,dt_g1_pre_topc,dt_k10_seqfunc,dt_k11_seq_1,dt_k12_mcart_1,dt_k12_seq_1,dt_k13_seq_1,dt_k14_seq_1,dt_k16_complex1,dt_k18_complex1,dt_k1_binop_1,dt_k1_domain_1,dt_k1_funct_1,dt_k1_metric_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_seq_1,dt_k1_seqfunc,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_newton,dt_k2_prepower,dt_k2_seq_1,dt_k2_series_1,dt_k2_tarski,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_prepower,dt_k3_real_1,dt_k3_tarski,dt_k3_topmetr,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_relset_1,dt_k4_tarski,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_seq_1,dt_k5_setfam_1,dt_k6_borsuk_1,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_k8_funct_2,dt_k8_seq_1,dt_k9_urysohn1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_seqfunc,dt_m1_subset_1,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_relset_1,existence_m1_seqfunc,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_xreal_0,fc12_membered,fc12_xreal_0,fc13_membered,fc13_xreal_0,fc14_finset_1,fc14_membered,fc14_xreal_0,fc15_membered,fc15_xreal_0,fc16_membered,fc16_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finset_1,fc1_nat_1,fc1_newton,fc1_pscomp_1,fc1_seq_1,fc1_struct_0,fc1_subset_1,fc1_xreal_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_borsuk_1,fc2_finseq_1,fc2_finset_1,fc2_membered,fc2_nat_1,fc2_newton,fc2_pscomp_1,fc2_subset_1,fc30_xreal_0,fc3_nat_1,fc3_subset_1,fc3_xreal_0,fc4_nat_1,fc4_subset_1,fc4_xreal_0,fc5_membered,fc5_seq_1,fc5_xreal_0,fc6_membered,fc6_seq_1,fc6_xreal_0,fc7_membered,fc7_topreal6,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_xreal_0,fraenkel_a_2_1_borsuk_1,fraenkel_a_3_0_borsuk_1,free_g1_pre_topc,involutiveness_k4_xcmplx_0,projectivity_k16_complex1,projectivity_k18_complex1,rc1_finseq_1,rc1_finset_1,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_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_k11_seq_1,redefinition_k12_mcart_1,redefinition_k13_seq_1,redefinition_k14_seq_1,redefinition_k18_complex1,redefinition_k1_domain_1,redefinition_k1_metric_1,redefinition_k1_seq_1,redefinition_k1_seqfunc,redefinition_k2_seq_1,redefinition_k3_prepower,redefinition_k3_real_1,redefinition_k4_real_1,redefinition_k4_relset_1,redefinition_k5_numbers,redefinition_k5_setfam_1,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_k8_seq_1,redefinition_k9_urysohn1,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_seqfunc__e3_19__nagata_2,s8_funct_2__e3_19_1__nagata_2,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t106_zfmisc_1,t11_nagata_2,t121_real_2,t12_seq_1,t13_seq_1,t13_series_1,t14_nagata_2,t1_arithm,t1_numerals,t1_real,t1_subset,t24_series_1,t24_topmetr,t28_nagata_1,t28_series_1,t29_nagata_1,t2_arithm,t2_real,t2_subset,t2_tarski,t33_jgraph_2,t34_partfun1,t35_partfun1,t3_arithm,t3_real,t3_subset,t4_arithm,t4_real,t4_subset,t5_arithm,t5_heine,t5_real,t5_subset,t66_xreal_1,t6_arithm,t6_boole,t6_real,t7_boole,t7_real,t83_topreal6,t8_boole,t8_real,t8_seq_1]), [file(nagata_2,t15_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_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_k2_tarski,axiom,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), []). fof(commutativity_k2_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), []). fof(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_xcmplx_0,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), []). fof(commutativity_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_1), []). fof(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(commutativity_k8_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,B) ) => k8_seq_1(A,B,C,D) = k8_seq_1(A,B,D,C) ) ), file(seq_1,k8_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_seqfunc,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_seqfunc(B,A,k1_numbers) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) ) => ( D = k10_seqfunc(A,B,C) <=> ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,D,E) = k2_seq_1(A,k1_numbers,k1_seqfunc(A,k1_numbers,B,E),C) ) ) ) ) ) ) ), file(seqfunc,d11_seqfunc), []). 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_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_prepower,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( B = k2_prepower(A) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_newton(A,C) ) ) ) ) ), file(prepower,d1_prepower), []). 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(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_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d6_seq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( C = k12_seq_1(A,B) <=> ( k1_relat_1(C) = k1_relat_1(A) & ! [D] : ( r2_hidden(D,k1_relat_1(C)) => k1_funct_1(C,D) = k3_xcmplx_0(B,k1_seq_1(A,D)) ) ) ) ) ) ) ), file(seq_1,d6_seq_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_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_k12_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_xreal_0(B) ) => ( v1_relat_1(k12_seq_1(A,B)) & v1_funct_1(k12_seq_1(A,B)) ) ) ), file(seq_1,k12_seq_1), []). fof(dt_k13_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_xreal_0(D) ) => ( v1_funct_1(k13_seq_1(A,B,C,D)) & m2_relset_1(k13_seq_1(A,B,C,D),A,k1_numbers) ) ) ), file(seq_1,k13_seq_1), []). fof(dt_k14_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_xreal_0(B) ) => ( v1_funct_1(k14_seq_1(A,B)) & v1_funct_2(k14_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k14_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k14_seq_1), []). fof(dt_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k16_complex1(A)) ) ), file(complex1,k16_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_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_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_metric_1,axiom,( ! [A,B,C,D,E] : ( ( v1_funct_1(C) & m1_relset_1(C,k2_zfmisc_1(A,B),k1_numbers) & m1_subset_1(D,A) & m1_subset_1(E,B) ) => m1_subset_1(k1_metric_1(A,B,C,D,E),k1_numbers) ) ), file(metric_1,k1_metric_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), []). fof(dt_k1_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => m1_subset_1(k1_seq_1(A,B),k1_numbers) ) ), file(seq_1,k1_seq_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_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_newton,axiom,( $true ), file(newton,k2_newton), []). 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_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_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_prepower,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => m1_subset_1(k3_prepower(A,B),k1_numbers) ) ), file(prepower,k3_prepower), []). 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_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_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), []). fof(dt_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), []). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), []). fof(dt_k4_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_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_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_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,B) ) => ( v1_funct_1(k8_seq_1(A,B,C,D)) & m2_relset_1(k8_seq_1(A,B,C,D),A,k1_numbers) ) ) ), file(seq_1,k8_seq_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_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_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_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_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_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), []). fof(fc10_membered,axiom,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), []). fof(fc10_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), []). fof(fc11_membered,axiom,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), []). fof(fc11_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), []). fof(fc12_membered,axiom,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), []). fof(fc12_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), []). fof(fc13_membered,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), []). fof(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_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_finset_1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_newton,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v4_ordinal2(B) ) => ( v1_xcmplx_0(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) ) ) ), file(newton,fc1_newton), []). 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(fc21_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), []). fof(fc22_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), []). fof(fc23_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), []). fof(fc24_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), []). fof(fc27_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_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_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_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(nat_1,fc2_nat_1), []). fof(fc2_newton,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_newton(A,B)) & v1_xcmplx_0(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) & ~ v3_xreal_0(k2_newton(A,B)) & v1_int_1(k2_newton(A,B)) ) ) ), file(newton,fc2_newton), []). 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(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(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_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_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(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_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), []). fof(fc5_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_seq_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k12_seq_1(B,A)) & v1_funct_1(k12_seq_1(B,A)) & v1_seq_1(k12_seq_1(B,A)) ) ) ), file(seq_1,fc6_seq_1), []). 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_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_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(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(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(projectivity_k16_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k16_complex1(k16_complex1(A)) = k16_complex1(A) ) ), file(complex1,k16_complex1), []). fof(projectivity_k18_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k18_complex1(k18_complex1(A)) = k18_complex1(A) ) ), file(complex1,k18_complex1), []). fof(rc1_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_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_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_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_k13_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_xreal_0(D) ) => k13_seq_1(A,B,C,D) = k12_seq_1(C,D) ) ), file(seq_1,k13_seq_1), []). fof(redefinition_k14_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_xreal_0(B) ) => k14_seq_1(A,B) = k12_seq_1(A,B) ) ), file(seq_1,k14_seq_1), []). 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_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_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => k1_seq_1(A,B) = k1_funct_1(A,B) ) ), file(seq_1,k1_seq_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_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_prepower,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k5_numbers) ) => k3_prepower(A,B) = k2_newton(A,B) ) ), file(prepower,k3_prepower), []). 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_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), []). fof(redefinition_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). 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_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_k8_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) & v1_funct_1(D) & m1_relset_1(D,A,B) ) => k8_seq_1(A,B,C,D) = k5_seq_1(C,D) ) ), file(seq_1,k8_seq_1), []). 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_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_seqfunc__e3_19__nagata_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_seqfunc(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),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) => k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,C,D) = k13_seq_1(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,B,D),k8_funct_2(k5_numbers,k1_numbers,k2_prepower(k6_real_1(1,2)),D)) ) ) ) ), file(nagata_2,s1_seqfunc__e3_19__nagata_2), []). fof(s8_funct_2__e3_19_1__nagata_2,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m2_subset_1(B,k1_numbers,k5_numbers) & 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] : ( 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) & ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(A)) => k1_funct_1(D,k4_tarski(E,F)) = k4_real_1(k8_funct_2(k5_numbers,k1_numbers,k2_prepower(k6_real_1(1,2)),B),k1_metric_1(u1_struct_0(A),u1_struct_0(A),C,E,F)) ) ) ) ) ), file(nagata_2,s8_funct_2__e3_19_1__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(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(t11_nagata_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_seqfunc(B,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) => ( ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [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) & k1_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers,B,C) = D & r1_nagata_1(u1_struct_0(A),D) ) ) & ! [C] : ( m1_subset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) => v1_series_1(k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,C)) ) ) => ! [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,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,C,D) = k2_series_1(k10_seqfunc(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),B,D)) ) => r1_nagata_1(u1_struct_0(A),C) ) ) ) ) ) ), file(nagata_2,t11_nagata_2), []). fof(t121_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ( r1_xreal_0(0,A) & r1_xreal_0(0,B) ) | ( r1_xreal_0(A,0) & r1_xreal_0(B,0) ) ) => r1_xreal_0(0,k3_xcmplx_0(A,B)) ) ) ) ), file(real_2,t121_real_2), []). fof(t12_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( A = k8_seq_1(k5_numbers,k1_numbers,B,C) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,D) = k4_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ), file(seq_1,t12_seq_1), []). fof(t13_seq_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( B = k14_seq_1(C,A) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,D) = k3_xcmplx_0(A,k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ), file(seq_1,t13_seq_1), []). fof(t13_series_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_series_1(B) => ( v1_series_1(k14_seq_1(B,A)) & k2_series_1(k14_seq_1(B,A)) = k3_xcmplx_0(A,k2_series_1(B)) ) ) ) ) ), file(series_1,t13_series_1), []). fof(t14_nagata_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_seqfunc(B,u1_struct_0(A),k1_numbers) => ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_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_seqfunc(u1_struct_0(A),k1_numbers,B,C) = D & v9_pscomp_1(D,A) & ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => r1_xreal_0(0,k8_funct_2(u1_struct_0(A),k1_numbers,D,E)) ) ) ) => ( ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ~ ( v1_series_1(C) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,k10_seqfunc(u1_struct_0(A),B,E),D),k8_funct_2(k5_numbers,k1_numbers,C,D)) ) ) ) ) | ! [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] : ( m1_subset_1(D,u1_struct_0(A)) => k8_funct_2(u1_struct_0(A),k1_numbers,C,D) = k2_series_1(k10_seqfunc(u1_struct_0(A),B,D)) ) => v9_pscomp_1(C,A) ) ) ) ) ) ) ), file(nagata_2,t14_nagata_2), []). fof(t1_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t24_series_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r1_xreal_0(0,k8_funct_2(k5_numbers,k1_numbers,A,C)) & r1_xreal_0(k8_funct_2(k5_numbers,k1_numbers,A,C),k8_funct_2(k5_numbers,k1_numbers,B,C)) ) ) & v1_series_1(B) ) => ( v1_series_1(A) & r1_xreal_0(k2_series_1(A),k2_series_1(B)) ) ) ) ) ), file(series_1,t24_series_1), []). fof(t24_topmetr,axiom,( u1_struct_0(k3_topmetr) = k1_numbers ), file(topmetr,t24_topmetr), []). 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_series_1,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ r1_xreal_0(1,k18_complex1(A)) => ( v1_series_1(k2_prepower(A)) & k2_series_1(k2_prepower(A)) = k7_xcmplx_0(1,k6_xcmplx_0(1,A)) ) ) ) ), file(series_1,t28_series_1), []). fof(t29_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) => r1_xreal_0(0,k1_metric_1(A,A,B,C,D)) ) ) ) ) ), file(nagata_1,t29_nagata_1), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). 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(t33_jgraph_2,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),u1_struct_0(k3_topmetr)) & m2_relset_1(B,u1_struct_0(A),u1_struct_0(k3_topmetr)) ) => ! [C] : ( v1_xreal_0(C) => ~ ( v5_pre_topc(B,A,k3_topmetr) & ! [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)) ) => ~ ( ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( v1_xreal_0(F) => ( k8_funct_2(u1_struct_0(A),u1_struct_0(k3_topmetr),B,E) = F => k8_funct_2(u1_struct_0(A),u1_struct_0(k3_topmetr),D,E) = k3_xcmplx_0(C,F) ) ) ) & v5_pre_topc(D,A,k3_topmetr) ) ) ) ) ) ) ), file(jgraph_2,t33_jgraph_2), []). fof(t34_partfun1,axiom,( ! [A,B,C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,B) ) => ! [D] : ( ( v1_funct_1(D) & m2_relset_1(D,A,B) ) => ( ( k4_relset_1(A,B,C) = k4_relset_1(A,B,D) & ! [E] : ( m1_subset_1(E,A) => ( r2_hidden(E,k4_relset_1(A,B,C)) => k1_funct_1(C,E) = k1_funct_1(D,E) ) ) ) => C = D ) ) ) ), file(partfun1,t34_partfun1), []). fof(t35_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(D) & m2_relset_1(D,k2_zfmisc_1(A,B),C) ) => ! [E] : ( ( v1_funct_1(E) & m2_relset_1(E,k2_zfmisc_1(A,B),C) ) => ( ( k4_relset_1(k2_zfmisc_1(A,B),C,D) = k4_relset_1(k2_zfmisc_1(A,B),C,E) & ! [F,G] : ( r2_hidden(k4_tarski(F,G),k4_relset_1(k2_zfmisc_1(A,B),C,D)) => k1_funct_1(D,k4_tarski(F,G)) = k1_funct_1(E,k4_tarski(F,G)) ) ) => D = E ) ) ) ), file(partfun1,t35_partfun1), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). fof(t3_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t5_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), []). fof(t5_heine,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_newton(A,B),0) ) ) ) ), file(heine,t5_heine), []). 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(t66_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(0,C) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t66_xreal_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(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t7_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), []). fof(t83_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(t8_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( ! [C] : ( r2_hidden(C,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,B,C) ) => A = B ) ) ) ), file(seq_1,t8_seq_1), []).