% Mizar problem: t11_heine,heine,209,58 fof(t11_heine,conjecture,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => v2_compts_1(k4_topmetr(A,B)) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_metric_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_finset_1,cc1_membered,cc1_pre_topc,cc1_relset_1,cc1_seqm_3,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,commutativity_k1_nat_1,commutativity_k2_xboole_0,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_metric_1,commutativity_k4_real_1,commutativity_k4_subset_1,connectedness_r1_xreal_0,d10_seqm_3,d13_metric_1,d14_metric_1,d1_absvalue,d1_funct_2,d1_heine,d1_metric_1,d1_rcomp_1,d1_tarski,d1_topmetr,d1_xreal_0,d3_rcomp_1,d3_seq_2,d3_tarski,d4_tarski,d4_topmetr,d5_funct_1,d5_topmetr,d6_topmetr,d7_seq_2,d8_seq_1,d8_seqm_3,d8_topmetr,d9_xcmplx_0,dt_g1_metric_1,dt_g1_pre_topc,dt_k12_seq_1,dt_k14_seq_1,dt_k16_complex1,dt_k17_complex1,dt_k18_complex1,dt_k18_seq_1,dt_k1_binop_1,dt_k1_funct_1,dt_k1_heine,dt_k1_metric_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_pcomps_1,dt_k1_rcomp_1,dt_k1_real_1,dt_k1_relat_1,dt_k1_seq_2,dt_k1_seq_4,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_metric_1,dt_k2_newton,dt_k2_real_1,dt_k2_relat_1,dt_k2_seq_1,dt_k2_seq_2,dt_k2_seqm_3,dt_k2_topmetr,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_newton,dt_k3_power,dt_k3_real_1,dt_k3_seqm_3,dt_k3_tarski,dt_k3_topmetr,dt_k3_xcmplx_0,dt_k4_metric_1,dt_k4_power,dt_k4_real_1,dt_k4_relset_1,dt_k4_subset_1,dt_k4_topmetr,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k5_real_1,dt_k5_relat_1,dt_k5_setfam_1,dt_k5_xcmplx_0,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_metric_1,dt_k7_xcmplx_0,dt_k8_funct_2,dt_k8_metric_1,dt_k9_metric_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_seqm_3,dt_m1_subset_1,dt_m1_topmetr,dt_m2_relset_1,dt_m2_subset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_seqm_3,existence_m1_subset_1,existence_m1_topmetr,existence_m2_relset_1,existence_m2_subset_1,fc10_membered,fc10_xreal_0,fc11_membered,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_finset_1,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finset_1,fc1_newton,fc1_ordinal2,fc1_subset_1,fc1_xreal_0,fc20_xreal_0,fc21_xreal_0,fc22_membered,fc22_xreal_0,fc23_membered,fc23_xreal_0,fc24_membered,fc24_xreal_0,fc25_membered,fc25_xreal_0,fc26_membered,fc26_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_membered,fc2_newton,fc2_subset_1,fc2_xreal_0,fc30_xreal_0,fc3_metric_1,fc3_xreal_0,fc4_metric_1,fc4_subset_1,fc4_xreal_0,fc5_membered,fc5_xreal_0,fc6_membered,fc6_xreal_0,fc7_membered,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_finset_1,fc9_membered,fc9_xreal_0,fraenkel_a_2_0_rcomp_1,fraenkel_a_2_1_heine,fraenkel_a_6_0_heine,free_g1_metric_1,free_g1_pre_topc,idempotence_k2_xboole_0,idempotence_k4_subset_1,involutiveness_k1_real_1,involutiveness_k2_real_1,involutiveness_k4_xcmplx_0,involutiveness_k5_xcmplx_0,l3_heine,l4_heine,projectivity_k16_complex1,projectivity_k17_complex1,projectivity_k18_complex1,rc1_arytm_3,rc1_finset_1,rc1_membered,rc1_metric_1,rc1_pre_topc,rc1_seqm_3,rc1_subset_1,rc1_xreal_0,rc2_metric_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_pre_topc,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_pre_topc,redefinition_k14_seq_1,redefinition_k17_complex1,redefinition_k18_complex1,redefinition_k1_metric_1,redefinition_k1_nat_1,redefinition_k1_pcomps_1,redefinition_k1_real_1,redefinition_k1_seq_4,redefinition_k2_real_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k2_seqm_3,redefinition_k3_newton,redefinition_k3_real_1,redefinition_k3_seqm_3,redefinition_k4_metric_1,redefinition_k4_power,redefinition_k4_real_1,redefinition_k4_relset_1,redefinition_k4_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k5_setfam_1,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,s1_nat_1__e18_11_1_1_2__heine,s1_nat_1__e22_11_1_1_2__heine,s2_recdef_1__e10_11_1_1_2__heine,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,t10_heine,t10_newton,t10_seq_4,t10_xreal_1,t119_xcmplx_1,t11_newton,t11_xreal_1,t121_real_2,t124_xreal_1,t125_real_2,t127_real_2,t12_absvalue,t12_metric_1,t138_complex1,t13_seq_1,t13_xboole_1,t142_complex1,t144_real_2,t14_finset_1,t14_rcomp_1,t14_topmetr,t151_real_2,t165_real_2,t173_real_2,t1_arithm,t1_boole,t1_heine,t1_numerals,t1_real,t1_subset,t1_xboole_1,t1_xcmplx_1,t217_xcmplx_1,t21_seq_2,t21_xreal_1,t22_seq_2,t22_xreal_1,t23_topmetr,t24_rcomp_1,t25_topmetr,t26_xcmplx_1,t26_xreal_1,t27_xcmplx_1,t29_nat_1,t2_arithm,t2_heine,t2_real,t2_subset,t2_tarski,t2_xreal_1,t30_pcomps_1,t31_seqm_3,t31_zfmisc_1,t37_zfmisc_1,t3_arithm,t3_real,t3_subset,t48_power,t4_arithm,t4_real,t4_subset,t50_xreal_1,t58_limfunc1,t5_arithm,t5_heine,t5_real,t5_subset,t61_limfunc1,t66_xcmplx_1,t6_arithm,t6_boole,t6_heine,t6_real,t70_xreal_1,t79_xcmplx_1,t7_boole,t7_real,t8_boole,t8_real,t8_xboole_1,t96_zfmisc_1,t9_heine]), [file(heine,t11_heine)]). fof(abstractness_v1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), []). fof(abstractness_v1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), []). fof(antisymmetry_r2_hidden,axiom,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), []). fof(cc10_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), []). fof(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(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_arytm_3,axiom,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), []). 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_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), []). fof(cc1_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_seqm_3,axiom,( ! [A] : ( m1_relset_1(A,k5_numbers,k1_numbers) => ( ( v1_funct_1(A) & v1_seqm_3(A) ) => ( v1_funct_1(A) & v1_seq_1(A) & v3_seqm_3(A) ) ) ) ), file(seqm_3,cc1_seqm_3), []). 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_arytm_3,axiom,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), []). 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_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc3_arytm_3,axiom,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). 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_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_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_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_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), []). fof(commutativity_k2_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_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_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k4_metric_1(A,C,B) ) ), file(metric_1,k4_metric_1), []). 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_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), []). fof(connectedness_r1_xreal_0,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), []). fof(d10_seqm_3,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) ) => ( m1_seqm_3(B,A) <=> ? [C] : ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & v1_seqm_3(C) & v7_seqm_3(C) & m2_relset_1(C,k5_numbers,k1_numbers) & B = k3_seqm_3(C,A) ) ) ) ) ), file(seqm_3,d10_seqm_3), []). fof(d13_metric_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ) => ( A = k7_metric_1 <=> ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => k1_metric_1(k1_numbers,k1_numbers,A,B,C) = k18_complex1(k5_real_1(B,C)) ) ) ) ) ), file(metric_1,d13_metric_1), []). fof(d14_metric_1,axiom,( k8_metric_1 = g1_metric_1(k1_numbers,k7_metric_1) ), file(metric_1,d14_metric_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_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_heine,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( C = k1_heine(A,B) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_power(B,k2_seq_1(k5_numbers,k1_numbers,A,D)) ) ) ) ) ) ), file(heine,d1_heine), []). fof(d1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k2_metric_1(A,B,C) = k1_metric_1(u1_struct_0(A),u1_struct_0(A),u1_metric_1(A),B,C) ) ) ) ), file(metric_1,d1_metric_1), []). fof(d1_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k1_rcomp_1(A,B) = a_2_0_rcomp_1(A,B) ) ) ), file(rcomp_1,d1_rcomp_1), []). fof(d1_tarski,axiom,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), []). fof(d1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) & l1_metric_1(B) ) => ( m1_topmetr(B,A) <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A)) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),C,D) = k1_binop_1(u1_metric_1(A),C,D) ) ) ) ) ) ) ), file(topmetr,d1_topmetr), []). fof(d1_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) <=> r2_hidden(A,k1_numbers) ) ), file(xreal_0,d1_xreal_0), []). fof(d3_rcomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_rcomp_1(A) <=> ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ~ ( r1_tarski(k2_relat_1(B),A) & ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ~ ( m1_seqm_3(C,B) & v4_seq_2(C) & r2_hidden(k2_seq_2(C),A) ) ) ) ) ) ) ), file(rcomp_1,d3_rcomp_1), []). fof(d3_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_seq_2(A) <=> ? [B] : ( v1_xreal_0(B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ r1_xreal_0(B,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ), file(seq_2,d3_seq_2), []). fof(d3_tarski,axiom,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), []). fof(d4_tarski,axiom,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), []). fof(d4_topmetr,axiom,( ! [A] : ( l1_metric_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_topmetr(B,A) <=> ! [C] : ~ ( r2_hidden(C,B) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,k1_numbers) => C != k9_metric_1(A,D,E) ) ) ) ) ) ) ), file(topmetr,d4_topmetr), []). 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_topmetr,axiom,( ! [A] : ( l1_metric_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( r1_topmetr(A,B) <=> r1_tarski(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) ) ), file(topmetr,d5_topmetr), []). fof(d6_topmetr,axiom,( ! [A] : ( l1_metric_1(A) => ( v2_topmetr(A) <=> v2_compts_1(k5_pcomps_1(A)) ) ) ), file(topmetr,d6_topmetr), []). fof(d7_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v4_seq_2(A) => ! [B] : ( v1_xreal_0(B) => ( B = k1_seq_2(A) <=> ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ? [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) & r1_xreal_0(D,E) & r1_xreal_0(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ), file(seq_2,d7_seq_2), []). fof(d8_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) ) => ( B = k18_seq_1(A) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_real_1(k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ), file(seq_1,d8_seq_1), []). fof(d8_seqm_3,axiom,( ! [A] : ( v1_relat_1(A) => ( v7_seqm_3(A) <=> r1_tarski(k2_relat_1(A),k5_numbers) ) ) ), file(seqm_3,d8_seqm_3), []). fof(d8_topmetr,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k4_topmetr(A,B) = k5_pcomps_1(k2_topmetr(A,B)) ) ) ), file(topmetr,d8_topmetr), []). fof(d9_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ), file(xcmplx_0,d9_xcmplx_0), []). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), []). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), []). fof(dt_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_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_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_k18_seq_1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_funct_1(k18_seq_1(A)) & v1_funct_2(k18_seq_1(A),k5_numbers,k1_numbers) & m2_relset_1(k18_seq_1(A),k5_numbers,k1_numbers) ) ) ), file(seq_1,k18_seq_1), []). fof(dt_k1_binop_1,axiom,( $true ), file(binop_1,k1_binop_1), []). fof(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), []). fof(dt_k1_heine,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_funct_1(k1_heine(A,B)) & v1_funct_2(k1_heine(A,B),k5_numbers,k1_numbers) & m2_relset_1(k1_heine(A,B),k5_numbers,k1_numbers) ) ) ), file(heine,k1_heine), []). 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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_pcomps_1,axiom,( ! [A] : m1_subset_1(k1_pcomps_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ), file(pcomps_1,k1_pcomps_1), []). fof(dt_k1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k1_rcomp_1), []). fof(dt_k1_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_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => v1_xreal_0(k1_seq_2(A)) ) ), file(seq_2,k1_seq_2), []). fof(dt_k1_seq_4,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k1_seq_4(A),k1_zfmisc_1(k1_numbers)) ) ), file(seq_4,k1_seq_4), []). 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_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k2_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k2_metric_1), []). fof(dt_k2_newton,axiom,( $true ), file(newton,k2_newton), []). fof(dt_k2_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k2_real_1(A),k1_numbers) ) ), file(real_1,k2_real_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_seq_2,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_seq_2(A),k1_numbers) ) ), file(seq_2,k2_seq_2), []). fof(dt_k2_seqm_3,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_seqm_3(A) & v7_seqm_3(A) & m1_relset_1(A,k5_numbers,k1_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k2_seqm_3(A,B),k1_numbers,k5_numbers) ) ), file(seqm_3,k2_seqm_3), []). fof(dt_k2_topmetr,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( ~ v3_struct_0(k2_topmetr(A,B)) & v1_metric_1(k2_topmetr(A,B)) & m1_topmetr(k2_topmetr(A,B),k8_metric_1) ) ) ), file(topmetr,k2_topmetr), []). 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_newton,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & v4_ordinal2(B) ) => m1_subset_1(k3_newton(A,B),k1_numbers) ) ), file(newton,k3_newton), []). fof(dt_k3_power,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k3_power(A,B)) ) ), file(power,k3_power), []). 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_seqm_3,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_seqm_3(A) & v7_seqm_3(A) & 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(k3_seqm_3(A,B)) & v1_funct_2(k3_seqm_3(A,B),k5_numbers,k1_numbers) & m2_relset_1(k3_seqm_3(A,B),k5_numbers,k1_numbers) ) ) ), file(seqm_3,k3_seqm_3), []). 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_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => m1_subset_1(k4_metric_1(A,B,C),k1_numbers) ) ), file(metric_1,k4_metric_1), []). fof(dt_k4_power,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_power(A,B),k1_numbers) ) ), file(power,k4_power), []). 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_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), []). fof(dt_k4_topmetr,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( ~ v3_struct_0(k4_topmetr(A,B)) & v1_pre_topc(k4_topmetr(A,B)) & m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ), file(topmetr,k4_topmetr), []). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), []). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), []). fof(dt_k5_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), []). fof(dt_k5_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_relat_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => v1_relat_1(k5_relat_1(A,B)) ) ), file(relat_1,k5_relat_1), []). fof(dt_k5_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_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k5_xcmplx_0(A)) ) ), file(xcmplx_0,k5_xcmplx_0), []). 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_metric_1,axiom, ( v1_funct_1(k7_metric_1) & v1_funct_2(k7_metric_1,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(k7_metric_1,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ), file(metric_1,k7_metric_1), []). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_metric_1,axiom, ( v1_metric_1(k8_metric_1) & l1_metric_1(k8_metric_1) ), file(metric_1,k8_metric_1), []). fof(dt_k9_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & v1_xreal_0(C) ) => m1_subset_1(k9_metric_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(metric_1,k9_metric_1), []). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), []). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), []). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), []). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_seqm_3,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( m1_seqm_3(B,A) => ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) ) ) ), file(seqm_3,m1_seqm_3), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_topmetr(B,A) => ( v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) & l1_metric_1(B) ) ) ) ), file(topmetr,m1_topmetr), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), []). fof(dt_u1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), []). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), []). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_seqm_3,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => ? [B] : m1_seqm_3(B,A) ) ), file(seqm_3,m1_seqm_3), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m1_topmetr,axiom,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ? [B] : m1_topmetr(B,A) ) ), file(topmetr,m1_topmetr), []). 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_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_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_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_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_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_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_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_ordinal2,axiom, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), []). 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_membered,axiom,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), []). 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_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(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_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(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(fc25_membered,axiom,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), []). fof(fc25_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & ~ v2_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc25_xreal_0), []). 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(fc26_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & ~ v3_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc26_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_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). 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_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc2_xreal_0), []). 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_metric_1,axiom, ( ~ v3_struct_0(k8_metric_1) & v1_metric_1(k8_metric_1) ), file(metric_1,fc3_metric_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_metric_1,axiom, ( ~ v3_struct_0(k8_metric_1) & v1_metric_1(k8_metric_1) & v6_metric_1(k8_metric_1) & v7_metric_1(k8_metric_1) & v8_metric_1(k8_metric_1) & v9_metric_1(k8_metric_1) ), file(metric_1,fc4_metric_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_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_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_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_2_0_rcomp_1,axiom,( ! [A,B,C] : ( ( v1_xreal_0(B) & v1_xreal_0(C) ) => ( r2_hidden(A,a_2_0_rcomp_1(B,C)) <=> ? [D] : ( m1_subset_1(D,k1_numbers) & A = D & r1_xreal_0(B,D) & r1_xreal_0(D,C) ) ) ) ), file(rcomp_1,a_2_0_rcomp_1), []). fof(fraenkel_a_2_1_heine,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_numbers) & m1_subset_1(C,k1_numbers) ) => ( r2_hidden(A,a_2_1_heine(B,C)) <=> ? [D] : ( m1_subset_1(D,k1_numbers) & A = D & r1_xreal_0(B,D) & r1_xreal_0(D,C) ) ) ) ), file(heine,a_2_1_heine), []). fof(fraenkel_a_6_0_heine,axiom,( ! [A,B,C,D,E,F,G] : ( ( m1_subset_1(B,k1_numbers) & m1_subset_1(C,k1_numbers) & v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) & v1_funct_1(E) & v1_funct_2(E,k5_numbers,k1_numbers) & v1_seqm_3(E) & v7_seqm_3(E) & m2_relset_1(E,k5_numbers,k1_numbers) & m2_subset_1(F,k1_numbers,k5_numbers) & m2_subset_1(G,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_6_0_heine(B,C,D,E,F,G)) <=> ? [H] : ( m1_subset_1(H,k1_numbers) & A = H & r1_xreal_0(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,D,k1_nat_1(k1_nat_1(G,1),F)),k4_real_1(k5_real_1(C,B),k2_real_1(k3_newton(2,k1_nat_1(k2_seqm_3(E,k1_nat_1(k1_nat_1(G,1),F)),1))))),H) & r1_xreal_0(H,k3_real_1(k2_seq_1(k5_numbers,k1_numbers,D,k1_nat_1(k1_nat_1(G,1),F)),k4_real_1(k5_real_1(C,B),k2_real_1(k3_newton(2,k1_nat_1(k2_seqm_3(E,k1_nat_1(k1_nat_1(G,1),F)),1)))))) ) ) ) ), file(heine,a_6_0_heine), []). fof(free_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), []). fof(free_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), []). fof(idempotence_k2_xboole_0,axiom,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), []). fof(idempotence_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), []). fof(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_k2_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k2_real_1(k2_real_1(A)) = A ) ), file(real_1,k2_real_1), []). fof(involutiveness_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), []). fof(involutiveness_k5_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => k5_xcmplx_0(k5_xcmplx_0(A)) = A ) ), file(xcmplx_0,k5_xcmplx_0), []). fof(l3_heine,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(0,A) & r1_xreal_0(k2_xcmplx_0(B,A),C) ) => r1_xreal_0(B,C) ) ) ) ) ), file(heine,l3_heine), []). fof(l4_heine,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(0,A) & r1_xreal_0(C,k6_xcmplx_0(B,A)) ) => r1_xreal_0(C,B) ) ) ) ) ), file(heine,l4_heine), []). 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(rc1_arytm_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), []). 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_metric_1,axiom,( ? [A] : ( l1_metric_1(A) & v1_metric_1(A) ) ), file(metric_1,rc1_metric_1), []). fof(rc1_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), []). fof(rc1_seqm_3,axiom,( ? [A] : ( m1_relset_1(A,k5_numbers,k1_numbers) & v1_relat_1(A) & v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_seq_1(A) & v1_seqm_3(A) & v3_seqm_3(A) & v7_seqm_3(A) ) ), file(seqm_3,rc1_seqm_3), []). 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_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_metric_1,axiom,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) ) ), file(metric_1,rc2_metric_1), []). fof(rc2_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), []). fof(rc2_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_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_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_metric_1,axiom,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) ) ), file(metric_1,rc3_metric_1), []). fof(rc3_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), []). fof(rc3_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_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), []). fof(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_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_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_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_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_seq_4,axiom,( ! [A] : ( v1_xreal_0(A) => k1_seq_4(A) = k1_tarski(A) ) ), file(seq_4,k1_seq_4), []). fof(redefinition_k2_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k2_real_1(A) = k5_xcmplx_0(A) ) ), file(real_1,k2_real_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_k2_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => k2_seq_2(A) = k1_seq_2(A) ) ), file(seq_2,k2_seq_2), []). fof(redefinition_k2_seqm_3,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_seqm_3(A) & v7_seqm_3(A) & m1_relset_1(A,k5_numbers,k1_numbers) & m1_subset_1(B,k5_numbers) ) => k2_seqm_3(A,B) = k1_funct_1(A,B) ) ), file(seqm_3,k2_seqm_3), []). fof(redefinition_k3_newton,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & v4_ordinal2(B) ) => k3_newton(A,B) = k2_newton(A,B) ) ), file(newton,k3_newton), []). 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_seqm_3,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_seqm_3(A) & v7_seqm_3(A) & 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) ) => k3_seqm_3(A,B) = k5_relat_1(A,B) ) ), file(seqm_3,k3_seqm_3), []). fof(redefinition_k4_metric_1,axiom,( ! [A,B,C] : ( ( v8_metric_1(A) & l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => k4_metric_1(A,B,C) = k2_metric_1(A,B,C) ) ), file(metric_1,k4_metric_1), []). fof(redefinition_k4_power,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_power(A,B) = k3_power(A,B) ) ), file(power,k4_power), []). 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_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), []). fof(redefinition_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_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_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_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_nat_1__e18_11_1_1_2__heine,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) & v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( ( r1_xreal_0(A,k5_real_1(k2_seq_1(k5_numbers,k1_numbers,C,0),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(0,1))))) & r1_xreal_0(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,C,0),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(0,1)))),B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(A,k5_real_1(k2_seq_1(k5_numbers,k1_numbers,C,D),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(D,1))))) & r1_xreal_0(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,C,D),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(D,1)))),B) ) => ( r1_xreal_0(A,k5_real_1(k2_seq_1(k5_numbers,k1_numbers,C,k1_nat_1(D,1)),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(k1_nat_1(D,1),1))))) & r1_xreal_0(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,C,k1_nat_1(D,1)),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(k1_nat_1(D,1),1)))),B) ) ) ) ) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(A,k5_real_1(k2_seq_1(k5_numbers,k1_numbers,C,D),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(D,1))))) & r1_xreal_0(k3_real_1(k2_seq_1(k5_numbers,k1_numbers,C,D),k6_real_1(k5_real_1(B,A),k3_newton(2,k1_nat_1(D,1)))),B) ) ) ) ) ), file(heine,s1_nat_1__e18_11_1_1_2__heine), []). fof(s1_nat_1__e22_11_1_1_2__heine,axiom,( ! [A,B,C,D,E,F] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & m1_subset_1(C,k1_numbers) & m1_subset_1(D,k1_numbers) & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) & v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_numbers) & m2_relset_1(F,k5_numbers,k1_numbers) ) => ( ( ! [G] : ( m1_subset_1(G,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) => ~ ( r1_tarski(G,E) & r1_tarski(k1_rcomp_1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,F,0),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(0,1)))),k3_real_1(k2_seq_1(k5_numbers,k1_numbers,F,0),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(0,1))))),k3_tarski(G)) & v1_finset_1(G) ) ) & ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ( ! [I] : ( m1_subset_1(I,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) => ~ ( r1_tarski(I,E) & r1_tarski(k1_rcomp_1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,F,H),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(H,1)))),k3_real_1(k2_seq_1(k5_numbers,k1_numbers,F,H),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(H,1))))),k3_tarski(I)) & v1_finset_1(I) ) ) => ! [J] : ( m1_subset_1(J,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) => ~ ( r1_tarski(J,E) & r1_tarski(k1_rcomp_1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,F,k1_nat_1(H,1)),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(k1_nat_1(H,1),1)))),k3_real_1(k2_seq_1(k5_numbers,k1_numbers,F,k1_nat_1(H,1)),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(k1_nat_1(H,1),1))))),k3_tarski(J)) & v1_finset_1(J) ) ) ) ) ) => ! [H] : ( m2_subset_1(H,k1_numbers,k5_numbers) => ! [K] : ( m1_subset_1(K,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) => ~ ( r1_tarski(K,E) & r1_tarski(k1_rcomp_1(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,F,H),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(H,1)))),k3_real_1(k2_seq_1(k5_numbers,k1_numbers,F,H),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(H,1))))),k3_tarski(K)) & v1_finset_1(K) ) ) ) ) ) ), file(heine,s1_nat_1__e22_11_1_1_2__heine), []). fof(s2_recdef_1__e10_11_1_1_2__heine,axiom,( ! [A,B,C,D,E] : ( ( v1_xreal_0(A) & v1_xreal_0(B) & m1_subset_1(C,k1_numbers) & m1_subset_1(D,k1_numbers) & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) ) => ( ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ! [G] : ( m1_subset_1(G,k1_numbers) => ? [H] : ( m1_subset_1(H,k1_numbers) & ( ! [I] : ( m1_subset_1(I,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) => ~ ( r1_tarski(I,E) & r1_tarski(k1_rcomp_1(k5_real_1(G,k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(F,1)))),G),k3_tarski(I)) & v1_finset_1(I) ) ) => H = k5_real_1(G,k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(F,2)))) ) & ( ? [I] : ( m1_subset_1(I,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) & r1_tarski(I,E) & r1_tarski(k1_rcomp_1(k5_real_1(G,k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(F,1)))),G),k3_tarski(I)) & v1_finset_1(I) ) => H = k3_real_1(G,k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(F,2)))) ) ) ) ) => ? [F] : ( v1_funct_1(F) & v1_funct_2(F,k5_numbers,k1_numbers) & m2_relset_1(F,k5_numbers,k1_numbers) & k8_funct_2(k5_numbers,k1_numbers,F,0) = k6_real_1(k3_real_1(C,D),2) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( ( ! [J] : ( m1_subset_1(J,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) => ~ ( r1_tarski(J,E) & r1_tarski(k1_rcomp_1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,F,G),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(G,1)))),k8_funct_2(k5_numbers,k1_numbers,F,G)),k3_tarski(J)) & v1_finset_1(J) ) ) => k8_funct_2(k5_numbers,k1_numbers,F,k1_nat_1(G,1)) = k5_real_1(k8_funct_2(k5_numbers,k1_numbers,F,G),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(G,2)))) ) & ( ? [J] : ( m1_subset_1(J,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k2_topmetr(A,B))))) & r1_tarski(J,E) & r1_tarski(k1_rcomp_1(k5_real_1(k8_funct_2(k5_numbers,k1_numbers,F,G),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(G,1)))),k8_funct_2(k5_numbers,k1_numbers,F,G)),k3_tarski(J)) & v1_finset_1(J) ) => k8_funct_2(k5_numbers,k1_numbers,F,k1_nat_1(G,1)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,F,G),k6_real_1(k5_real_1(D,C),k3_newton(2,k1_nat_1(G,2)))) ) ) ) ) ) ) ), file(heine,s2_recdef_1__e10_11_1_1_2__heine), []). 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(t10_heine,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_finset_1(u1_struct_0(A)) => v2_compts_1(A) ) ) ), file(heine,t10_heine), []). fof(t10_newton,axiom,( ! [A] : ( v1_xreal_0(A) => k2_newton(A,1) = A ) ), file(newton,t10_newton), []). fof(t10_seq_4,axiom,( ! [A] : ( v1_xreal_0(A) => ~ ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(B,A) ) ) ), file(seq_4,t10_seq_4), []). 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(t119_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k2_xcmplx_0(k7_xcmplx_0(A,k3_xcmplx_0(2,B)),k7_xcmplx_0(A,k3_xcmplx_0(2,B))) = k7_xcmplx_0(A,B) ) ) ), file(xcmplx_1,t119_xcmplx_1), []). fof(t11_newton,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_xreal_0(B) => k2_newton(B,k2_xcmplx_0(A,1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) ), file(newton,t11_newton), []). fof(t11_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(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), []). 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(t124_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k5_xcmplx_0(A),0) ) & ~ ( ~ r1_xreal_0(k5_xcmplx_0(A),0) & r1_xreal_0(A,0) ) ) ) ), file(xreal_1,t124_xreal_1), []). fof(t125_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ( r1_xreal_0(A,0) & r1_xreal_0(B,0) ) | ( r1_xreal_0(0,A) & r1_xreal_0(0,B) ) ) => r1_xreal_0(0,k7_xcmplx_0(A,B)) ) ) ) ), file(real_2,t125_real_2), []). fof(t127_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) ) | ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(0,B) ) ) & r1_xreal_0(k7_xcmplx_0(A,B),0) ) ) ) ), file(real_2,t127_real_2), []). fof(t12_absvalue,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(k4_xcmplx_0(A),B) & r1_xreal_0(B,A) ) <=> r1_xreal_0(k18_complex1(B),A) ) ) ) ), file(absvalue,t12_absvalue), []). fof(t12_metric_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( l1_metric_1(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => ( r2_hidden(D,k9_metric_1(B,C,A)) <=> ( ~ v3_struct_0(B) & ~ r1_xreal_0(A,k2_metric_1(B,C,D)) ) ) ) ) ) ) ), file(metric_1,t12_metric_1), []). fof(t138_complex1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k4_xcmplx_0(A)) = k17_complex1(A) ) ), file(complex1,t138_complex1), []). 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_xboole_1,axiom,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) ) => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) ), file(xboole_1,t13_xboole_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(t144_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,1) ) | ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(1,B) ) ) & r1_xreal_0(k3_xcmplx_0(A,B),A) ) ) ) ), file(real_2,t144_real_2), []). 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(t14_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => k1_rcomp_1(A,A) = k1_seq_4(A) ) ), file(rcomp_1,t14_rcomp_1), []). fof(t14_topmetr,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => u1_struct_0(k2_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ), file(topmetr,t14_topmetr), []). fof(t151_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ ( r1_xreal_0(A,0) & r1_xreal_0(0,B) ) & ~ r1_xreal_0(B,A) & r1_xreal_0(k7_xcmplx_0(1,A),k7_xcmplx_0(1,B)) ) ) ) ), file(real_2,t151_real_2), []). fof(t165_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,k6_xcmplx_0(B,C)) => r1_xreal_0(C,k6_xcmplx_0(B,A)) ) & ( r1_xreal_0(k6_xcmplx_0(B,C),A) => r1_xreal_0(k6_xcmplx_0(B,A),C) ) ) ) ) ) ), file(real_2,t165_real_2), []). fof(t173_real_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ~ r1_xreal_0(0,A) => ( ~ r1_xreal_0(B,k2_xcmplx_0(A,B)) & ~ r1_xreal_0(k6_xcmplx_0(B,A),B) ) ) & ~ ( ~ ( r1_xreal_0(B,k2_xcmplx_0(A,B)) & r1_xreal_0(k6_xcmplx_0(B,A),B) ) & r1_xreal_0(0,A) ) ) ) ) ), file(real_2,t173_real_2), []). 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_heine,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_topmetr(C,k8_metric_1) => ! [D] : ( m1_subset_1(D,u1_struct_0(C)) => ! [E] : ( m1_subset_1(E,u1_struct_0(C)) => ( ( A = D & B = E ) => k4_metric_1(C,D,E) = k18_complex1(k6_xcmplx_0(A,B)) ) ) ) ) ) ) ), file(heine,t1_heine), []). fof(t1_numerals,axiom,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), []). fof(t1_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), []). fof(t1_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t1_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), []). fof(t1_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k2_xcmplx_0(A,k2_xcmplx_0(B,C)) = k2_xcmplx_0(k2_xcmplx_0(A,B),C) ) ) ) ), file(xcmplx_1,t1_xcmplx_1), []). fof(t217_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(1,A) = k5_xcmplx_0(A) ) ), file(xcmplx_1,t217_xcmplx_1), []). fof(t21_seq_2,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) ) => ( v4_seq_2(B) => v4_seq_2(k14_seq_1(B,A)) ) ) ) ), file(seq_2,t21_seq_2), []). fof(t21_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(k2_xcmplx_0(A,B),C) <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ), file(xreal_1,t21_xreal_1), []). fof(t22_seq_2,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) ) => ( v4_seq_2(B) => k2_seq_2(k14_seq_1(B,A)) = k3_xcmplx_0(A,k2_seq_2(B)) ) ) ) ), file(seq_2,t22_seq_2), []). fof(t22_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,k2_xcmplx_0(B,C)) <=> r1_xreal_0(k6_xcmplx_0(A,B),C) ) ) ) ) ), file(xreal_1,t22_xreal_1), []). fof(t23_topmetr,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ( v2_topmetr(A) <=> ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ~ ( v1_topmetr(B,A) & r1_topmetr(A,B) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ~ ( r1_tarski(C,B) & r1_topmetr(A,C) & v1_finset_1(C) ) ) ) ) ) ) ), file(topmetr,t23_topmetr), []). fof(t24_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => v1_rcomp_1(k1_rcomp_1(A,B)) ) ) ), file(rcomp_1,t24_rcomp_1), []). fof(t25_topmetr,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => u1_struct_0(k4_topmetr(A,B)) = k1_rcomp_1(A,B) ) ) ) ), file(topmetr,t25_topmetr), []). fof(t26_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => A = k6_xcmplx_0(k2_xcmplx_0(A,B),B) ) ) ), file(xcmplx_1,t26_xcmplx_1), []). fof(t26_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k4_xcmplx_0(B),k4_xcmplx_0(A)) ) ) ) ), file(xreal_1,t26_xreal_1), []). fof(t27_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => A = k2_xcmplx_0(k6_xcmplx_0(A,B),B) ) ) ), file(xcmplx_1,t27_xcmplx_1), []). fof(t29_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_heine,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) ) => k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_rcomp_1(B,C)) = k1_rcomp_1(A,C) ) ) ) ) ), file(heine,t2_heine), []). 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_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( v1_xreal_0(D) => ~ ( v9_metric_1(A) & r2_hidden(B,k9_metric_1(A,C,D)) & ! [E] : ( m1_subset_1(E,k1_numbers) => ~ ( ~ r1_xreal_0(E,0) & r1_tarski(k9_metric_1(A,B,E),k9_metric_1(A,C,D)) ) ) ) ) ) ) ) ), file(pcomps_1,t30_pcomps_1), []). fof(t31_seqm_3,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) & v1_seqm_3(B) & v7_seqm_3(B) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k1_funct_1(k5_relat_1(B,A),C) = k2_seq_1(k5_numbers,k1_numbers,A,k2_seq_1(k5_numbers,k1_numbers,B,C)) ) ) ) ), file(seqm_3,t31_seqm_3), []). fof(t31_zfmisc_1,axiom,( ! [A] : k3_tarski(k1_tarski(A)) = A ), file(zfmisc_1,t31_zfmisc_1), []). fof(t37_zfmisc_1,axiom,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_1), []). 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(t48_power,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( A != 0 => k3_power(A,B) = k2_newton(A,B) ) ) ) ), file(power,t48_power), []). 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(t50_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => r1_xreal_0(0,k6_xcmplx_0(B,A)) ) ) ) ), file(xreal_1,t50_xreal_1), []). fof(t58_limfunc1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_seqm_3(A) => ( v1_seq_2(A) | v1_limfunc1(A) ) ) ) ), file(limfunc1,t58_limfunc1), []). 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(t61_limfunc1,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( ( v1_limfunc1(A) | v2_limfunc1(A) ) => ( v4_seq_2(k18_seq_1(A)) & k2_seq_2(k18_seq_1(A)) = 0 ) ) ) ), file(limfunc1,t61_limfunc1), []). fof(t66_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(k7_xcmplx_0(A,2),k7_xcmplx_0(A,2)) = A ) ), file(xcmplx_1,t66_xcmplx_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_heine,axiom,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( ( v1_seqm_3(B) & r1_tarski(k2_relat_1(B),k5_numbers) ) => r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,B,A)) ) ) ) ), file(heine,t6_heine), []). 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(t70_xreal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(C,B) & r1_xreal_0(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ), file(xreal_1,t70_xreal_1), []). fof(t79_xcmplx_1,axiom,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k7_xcmplx_0(A,k3_xcmplx_0(B,C)) = k7_xcmplx_0(k7_xcmplx_0(A,B),C) ) ) ) ), file(xcmplx_1,t79_xcmplx_1), []). 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(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t8_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), []). fof(t8_xboole_1,axiom,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), []). fof(t96_zfmisc_1,axiom,( ! [A,B] : k3_tarski(k2_xboole_0(A,B)) = k2_xboole_0(k3_tarski(A),k3_tarski(B)) ), file(zfmisc_1,t96_zfmisc_1), []). fof(t9_heine,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_limfunc1(A) => v1_limfunc1(k1_heine(A,2)) ) ) ), file(heine,t9_heine), []).