% Mizar problem: t25_topalg_5,topalg_5,3344,55 fof(t25_topalg_5,conjecture,( ! [A] : ( v1_int_1(A) => ! [B] : ( m1_borsuk_2(B,k3_topalg_2,k4_toprealb(0),k4_toprealb(A)) => k1_funct_1(k7_topalg_5,A) = k6_eqrel_1(k1_topalg_1(k8_toprealb(2),k9_toprealb),k2_topalg_1(k8_toprealb(2),k9_toprealb),k4_borsuk_1(k5_topmetr,k3_topalg_2,k8_toprealb(2),B,k12_toprealb)) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_group_1,abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc10_yellow13,cc11_membered,cc11_topreal6,cc11_yellow13,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_borsuk_2,cc1_funct_1,cc1_membered,cc1_nat_1,cc1_pre_topc,cc1_realset1,cc1_relset_1,cc1_topalg_1,cc1_topalg_3,cc1_topalg_5,cc1_topreal6,cc1_toprealb,cc1_tops_1,cc1_xreal_0,cc1_yellow13,cc20_membered,cc2_borsuk_2,cc2_funct_1,cc2_int_1,cc2_membered,cc2_nat_1,cc2_topalg_3,cc2_topreal6,cc2_tops_1,cc2_xreal_0,cc2_yellow13,cc3_borsuk_2,cc3_borsuk_5,cc3_int_1,cc3_membered,cc3_nat_1,cc3_topalg_2,cc3_topalg_3,cc3_tops_1,cc3_xreal_0,cc3_yellow13,cc4_borsuk_2,cc4_int_1,cc4_membered,cc4_topalg_2,cc4_topreal6,cc4_tops_1,cc4_xreal_0,cc4_yellow13,cc5_topalg_2,cc5_topreal6,cc5_tops_1,cc5_xreal_0,cc5_yellow13,cc6_membered,cc6_topalg_2,cc6_topreal6,cc6_tops_1,cc6_xreal_0,cc6_yellow13,cc7_topreal6,cc7_xreal_0,cc7_yellow13,cc8_topreal6,cc8_xreal_0,cc8_yellow13,cc9_membered,cc9_topreal6,cc9_yellow13,commutativity_k2_tarski,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,commutativity_k4_real_1,d11_toprealb,d1_binop_1,d2_toprealb,d4_borsuk_2,d5_tarski,d5_topalg_2,d5_topalg_5,d7_borsuk_2,d8_toprealb,de_c4__topalg_5,de_c5__topalg_5,dt_c4__topalg_5,dt_c5__topalg_5,dt_g1_group_1,dt_g1_pre_topc,dt_k12_toprealb,dt_k15_euclid,dt_k19_sin_cos,dt_k1_binop_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_tarski,dt_k1_topalg_1,dt_k1_topalg_5,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k20_sin_cos,dt_k22_borsuk_1,dt_k22_sin_cos,dt_k23_euclid,dt_k23_sin_cos,dt_k2_tarski,dt_k2_topalg_1,dt_k2_topalg_5,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k31_sin_cos,dt_k32_sin_cos,dt_k3_gr_cy_1,dt_k3_topalg_1,dt_k3_topalg_2,dt_k3_topmetr,dt_k3_xcmplx_0,dt_k4_borsuk_1,dt_k4_real_1,dt_k4_tarski,dt_k4_topalg_2,dt_k4_toprealb,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_ordinal2,dt_k5_relat_1,dt_k5_topalg_2,dt_k5_topalg_5,dt_k5_topmetr,dt_k6_borsuk_1,dt_k6_eqrel_1,dt_k6_topalg_5,dt_k6_xcmplx_0,dt_k7_funct_2,dt_k7_topalg_5,dt_k8_funct_2,dt_k8_toprealb,dt_k9_toprealb,dt_l1_group_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_borsuk_2,dt_m1_borsuk_6,dt_m1_pre_topc,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_group_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_group_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_borsuk_2,existence_m1_borsuk_6,existence_m1_pre_topc,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,fc10_borsuk_5,fc10_membered,fc10_xreal_0,fc11_borsuk_5,fc11_membered,fc11_xreal_0,fc12_membered,fc12_xreal_0,fc13_membered,fc13_xreal_0,fc14_membered,fc14_xreal_0,fc15_membered,fc15_xreal_0,fc16_membered,fc16_xreal_0,fc17_xreal_0,fc18_borsuk_5,fc18_xreal_0,fc19_xreal_0,fc1_borsuk_5,fc1_funct_1,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_topalg_1,fc1_topalg_3,fc1_topalg_5,fc1_xboole_0,fc1_xreal_0,fc1_yellow13,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_toprealb,fc23_xreal_0,fc24_toprealb,fc24_xreal_0,fc28_toprealb,fc29_toprealb,fc2_borsuk_1,fc2_int_1,fc2_membered,fc2_nat_1,fc2_rcomp_3,fc2_subset_1,fc2_topalg_1,fc2_topalg_2,fc2_topalg_3,fc2_topalg_5,fc2_yellow13,fc3_int_1,fc3_nat_1,fc3_realset1,fc3_sin_cos,fc3_subset_1,fc3_topalg_1,fc3_topalg_2,fc3_xreal_0,fc4_borsuk_2,fc4_int_1,fc4_nat_1,fc4_sin_cos,fc4_subset_1,fc4_topalg_1,fc4_topalg_5,fc4_xreal_0,fc5_borsuk_1,fc5_borsuk_2,fc5_borsuk_5,fc5_int_1,fc5_membered,fc5_xreal_0,fc5_yellow12,fc6_borsuk_2,fc6_borsuk_5,fc6_int_1,fc6_membered,fc6_topalg_1,fc6_topalg_5,fc6_yellow12,fc7_borsuk_2,fc7_borsuk_5,fc7_int_1,fc7_membered,fc7_topreal6,fc7_xreal_0,fc8_borsuk_5,fc8_int_1,fc8_membered,fc8_xreal_0,fc9_borsuk_5,fc9_int_1,fc9_membered,fc9_xreal_0,free_g1_group_1,free_g1_pre_topc,involutiveness_k4_xcmplx_0,l5_topalg_5,rc1_borsuk_2,rc1_borsuk_5,rc1_funct_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_realset1,rc1_subset_1,rc1_topreal6,rc1_toprealb,rc1_tops_1,rc1_xboole_0,rc1_xreal_0,rc2_borsuk_1,rc2_borsuk_2,rc2_borsuk_5,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_realset1,rc2_subset_1,rc2_tops_1,rc2_xboole_0,rc2_xreal_0,rc3_borsuk_2,rc3_borsuk_5,rc3_funct_1,rc3_nat_1,rc3_pre_topc,rc3_struct_0,rc3_topalg_2,rc3_tops_1,rc3_xreal_0,rc4_pre_topc,rc4_rcomp_3,rc4_topalg_2,rc4_tops_1,rc4_xreal_0,rc5_pre_topc,rc5_rcomp_3,rc5_struct_0,rc5_tops_1,rc6_pre_topc,rc7_pre_topc,redefinition_k20_sin_cos,redefinition_k23_sin_cos,redefinition_k2_topalg_5,redefinition_k32_sin_cos,redefinition_k3_topalg_2,redefinition_k4_borsuk_1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_k5_topalg_2,redefinition_k5_topmetr,redefinition_k6_topalg_5,redefinition_k7_funct_2,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_r4_borsuk_2,reflexivity_r1_tarski,reflexivity_r4_borsuk_2,spc0_boole,spc1_boole,spc1_numerals,spc2_boole,spc2_numerals,symmetry_r3_borsuk_2,symmetry_r4_borsuk_2,t10_complex2,t1_arithm,t1_numerals,t1_subset,t20_topalg_5,t21_funct_2,t2_arithm,t2_subset,t33_toprealb,t34_sin_cos,t3_arithm,t3_subset,t3_topalg_3,t47_topalg_1,t4_arithm,t4_subset,t58_tops_2,t5_subset,t6_boole,t7_boole,t8_boole,t9_complex2]), [file(topalg_5,t25_topalg_5)]). fof(abstractness_v1_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( v1_group_1(A) => A = g1_group_1(u1_struct_0(A),u1_group_1(A)) ) ) ), file(group_1,v1_group_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(cc10_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v3_compts_1(A) ) => ( v2_pre_topc(A) & v1_urysohn1(A) ) ) ) ), file(yellow13,cc10_yellow13), []). fof(cc11_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), []). fof(cc11_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_finset_1(A) => ( v1_membered(A) & v2_membered(A) & v1_rcomp_1(A) ) ) ) ), file(topreal6,cc11_topreal6), []). fof(cc11_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v1_urysohn1(A) ) => ( v2_pre_topc(A) & v2_t_0topsp(A) ) ) ) ), file(yellow13,cc11_yellow13), []). fof(cc12_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), []). fof(cc13_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), []). fof(cc14_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), []). fof(cc15_membered,axiom,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), []). fof(cc16_membered,axiom,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), []). fof(cc17_membered,axiom,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), []). fof(cc18_membered,axiom,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), []). fof(cc19_membered,axiom,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), []). fof(cc1_borsuk_2,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ( v1_xreal_0(A) & v1_xcmplx_0(A) ) ) ), file(borsuk_2,cc1_borsuk_2), []). fof(cc1_funct_1,axiom,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), []). fof(cc1_membered,axiom,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), []). fof(cc1_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), []). fof(cc1_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), []). fof(cc1_realset1,axiom,( ! [A] : ( ~ v1_realset1(A) => ~ v1_xboole_0(A) ) ), file(realset1,cc1_realset1), []). fof(cc1_relset_1,axiom,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), []). fof(cc1_topalg_1,axiom,( ! [A,B,C,D,E] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) & m1_borsuk_2(D,k15_euclid(A),B,C) & m1_borsuk_2(E,k15_euclid(A),B,C) ) => ! [F] : ( m1_borsuk_6(F,k15_euclid(A),B,C,D,E) => ( v1_relat_1(F) & v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),k15_euclid(A)) ) ) ) ), file(topalg_1,cc1_topalg_1), []). fof(cc1_topalg_3,axiom,( ! [A] : ( l1_pre_topc(A) => ( v3_struct_0(A) => v1_connsp_1(A) ) ) ), file(topalg_3,cc1_topalg_3), []). fof(cc1_topalg_5,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k3_gr_cy_1)) => ( v1_xreal_0(A) & v1_xcmplx_0(A) & v1_int_1(A) ) ) ), file(topalg_5,cc1_topalg_5), []). fof(cc1_topreal6,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_connsp_1(B,A) ) ) ) ), file(topreal6,cc1_topreal6), []). fof(cc1_toprealb,axiom,( ! [A] : ( m1_pre_topc(A,k15_euclid(2)) => ( v1_toprealb(A) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) & v10_pscomp_1(A) & v1_borsuk_2(A) & v1_monoid_0(A) & v2_monoid_0(A) ) ) ) ), file(toprealb,cc1_toprealb), []). fof(cc1_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => ( v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ) ) ), file(tops_1,cc1_tops_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(cc1_yellow13,axiom,( ! [A] : ( v1_realset1(A) => v1_finset_1(A) ) ), file(yellow13,cc1_yellow13), []). fof(cc20_membered,axiom,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), []). fof(cc2_borsuk_2,axiom,( ! [A,B,C] : ( ( v1_borsuk_2(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ! [D] : ( m1_borsuk_2(D,A,B,C) => ( v1_relat_1(D) & v5_pre_topc(D,k5_topmetr,A) ) ) ) ), file(borsuk_2,cc2_borsuk_2), []). fof(cc2_funct_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), []). fof(cc2_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_nat_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), []). fof(cc2_topalg_3,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v3_realset2(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & v1_connsp_1(A) ) ) ) ), file(topalg_3,cc2_topalg_3), []). fof(cc2_topreal6,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v6_compts_1(B,A) ) ) ) ), file(topreal6,cc2_topreal6), []). fof(cc2_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc2_tops_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc2_yellow13,axiom,( ! [A] : ( l1_struct_0(A) => ( v3_realset2(A) => v6_group_1(A) ) ) ), file(yellow13,cc2_yellow13), []). fof(cc3_borsuk_2,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_connsp_1(A) ) ) ) ), file(borsuk_2,cc3_borsuk_2), []). fof(cc3_borsuk_5,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v1_rat_1(A) ) => ( ~ v1_xboole_0(A) & v1_xreal_0(A) & v1_xcmplx_0(A) ) ) ), file(borsuk_5,cc3_borsuk_5), []). fof(cc3_int_1,axiom,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), []). fof(cc3_membered,axiom,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), []). fof(cc3_nat_1,axiom,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), []). fof(cc3_topalg_2,axiom,( ! [A] : ( m1_pre_topc(A,k3_topmetr) => ( v2_pre_topc(A) & v1_borsuk_6(A) ) ) ), file(topalg_2,cc3_topalg_2), []). fof(cc3_topalg_3,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ! [C] : ( m1_borsuk_2(C,A,B,B) => ( v1_relat_1(C) & v5_pre_topc(C,k5_topmetr,A) ) ) ) ), file(topalg_3,cc3_topalg_3), []). fof(cc3_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v3_tops_1(B,A) ) ) ) ), file(tops_1,cc3_tops_1), []). fof(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). fof(cc3_yellow13,axiom,( ! [A] : ( l1_struct_0(A) => ( ~ v3_realset2(A) => ~ v3_struct_0(A) ) ) ), file(yellow13,cc3_yellow13), []). fof(cc4_borsuk_2,axiom,( ! [A] : ( l1_pre_topc(A) => ( v3_struct_0(A) => v2_t_0topsp(A) ) ) ), file(borsuk_2,cc4_borsuk_2), []). fof(cc4_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), []). fof(cc4_membered,axiom,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), []). fof(cc4_topalg_2,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( v1_xboole_0(A) => v2_topalg_2(A) ) ) ), file(topalg_2,cc4_topalg_2), []). fof(cc4_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v1_sppol_1(A) => ~ v1_xboole_0(A) ) ) ), file(topreal6,cc4_topreal6), []). fof(cc4_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_tops_1(B,A) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc4_tops_1), []). fof(cc4_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), []). fof(cc4_yellow13,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v4_pre_topc(B,A) ) ) ) ), file(yellow13,cc4_yellow13), []). fof(cc5_topalg_2,axiom,( ! [A] : ( m1_pre_topc(A,k3_topalg_2) => ( ( ~ v3_struct_0(A) & v3_topalg_2(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & v1_borsuk_6(A) & v1_connsp_1(A) ) ) ) ), file(topalg_2,cc5_topalg_2), []). fof(cc5_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ~ v2_sppol_1(A) => ~ v1_xboole_0(A) ) ) ), file(topreal6,cc5_topreal6), []). fof(cc5_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v4_pre_topc(B,A) & v2_tops_1(B,A) ) => ( v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ) ) ), file(tops_1,cc5_tops_1), []). fof(cc5_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), []). fof(cc5_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & v6_group_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_tdlat_3(A) & v3_tdlat_3(A) & v4_tdlat_3(A) & v5_tdlat_3(A) ) ) ) ), file(yellow13,cc5_yellow13), []). fof(cc6_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), []). fof(cc6_topalg_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v3_topalg_2(A) & m1_pre_topc(A,k3_topalg_2) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ! [F] : ( m1_borsuk_6(F,A,B,C,D,E) => ( v1_relat_1(F) & v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),A) ) ) ) ), file(topalg_2,cc6_topalg_2), []). fof(cc6_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v2_topreal4(A,k15_euclid(2)) => ( v3_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) ) ) ) ), file(topreal6,cc6_topreal6), []). fof(cc6_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v3_pre_topc(B,A) & v3_tops_1(B,A) ) => ( v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ) ) ), file(tops_1,cc6_tops_1), []). fof(cc6_xreal_0,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), []). fof(cc6_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v6_group_1(A) ) => ( v2_pre_topc(A) & v2_compts_1(A) ) ) ) ), file(yellow13,cc6_yellow13), []). fof(cc7_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ( v3_pre_topc(A,k15_euclid(2)) & v2_connsp_1(A,k15_euclid(2)) ) => v2_topreal4(A,k15_euclid(2)) ) ) ), file(topreal6,cc7_topreal6), []). fof(cc7_xreal_0,axiom,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), []). fof(cc7_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_tdlat_3(A) ) => ( v2_pre_topc(A) & v1_urysohn1(A) & v3_compts_1(A) & v4_compts_1(A) & v5_compts_1(A) ) ) ) ), file(yellow13,cc7_yellow13), []). fof(cc8_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_xboole_0(A) => v1_sppol_1(A) ) ) ), file(topreal6,cc8_topreal6), []). fof(cc8_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), []). fof(cc8_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & v5_compts_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v4_compts_1(A) ) ) ) ), file(yellow13,cc8_yellow13), []). fof(cc9_membered,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), []). fof(cc9_topreal6,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_xboole_0(A) => v2_sppol_1(A) ) ) ), file(topreal6,cc9_topreal6), []). fof(cc9_yellow13,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( v2_pre_topc(A) & v1_urysohn1(A) & v4_compts_1(A) ) => ( v2_pre_topc(A) & v3_compts_1(A) ) ) ) ), file(yellow13,cc9_yellow13), []). fof(commutativity_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_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(d11_toprealb,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) & m2_relset_1(A,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) ) => ( A = k12_toprealb <=> ! [B] : ( v1_xreal_0(B) => k1_funct_1(A,B) = k23_euclid(k22_sin_cos(k3_xcmplx_0(k4_real_1(2,k32_sin_cos),B)),k19_sin_cos(k3_xcmplx_0(k4_real_1(2,k32_sin_cos),B))) ) ) ) ), file(toprealb,d11_toprealb), []). 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(d2_toprealb,axiom,( ! [A] : ( v1_xreal_0(A) => k4_toprealb(A) = A ) ), file(toprealb,d2_toprealb), []). fof(d4_borsuk_2,axiom,( ! [A] : ( ( v1_borsuk_2(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) => ( m1_borsuk_2(D,A,B,C) <=> ( v5_pre_topc(D,k5_topmetr,A) & k1_funct_1(D,0) = B & k1_funct_1(D,1) = C ) ) ) ) ) ) ), file(borsuk_2,d4_borsuk_2), []). fof(d5_tarski,axiom,( ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ), file(tarski,d5_tarski), []). fof(d5_topalg_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v3_topalg_2(A) & m1_pre_topc(A,k3_topalg_2) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_borsuk_2(D,A,B,C) => ! [E] : ( m1_borsuk_2(E,A,B,C) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) & m2_relset_1(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) ) => ( F = k4_topalg_2(A,B,C,D,E) <=> ! [G] : ( m1_subset_1(G,u1_struct_0(k5_topmetr)) => ! [H] : ( m1_subset_1(H,u1_struct_0(k5_topmetr)) => k1_binop_1(F,G,H) = k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(1,H),k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),D,G)),k3_xcmplx_0(H,k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),E,G))) ) ) ) ) ) ) ) ) ) ), file(topalg_2,d5_topalg_2), []). fof(d5_topalg_5,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(2),k9_toprealb))) & m2_relset_1(A,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(2),k9_toprealb))) ) => ( A = k7_topalg_5 <=> ! [B] : ( v1_int_1(B) => k1_funct_1(A,B) = k6_eqrel_1(k1_topalg_1(k8_toprealb(2),k9_toprealb),k2_topalg_1(k8_toprealb(2),k9_toprealb),k6_topalg_5(B)) ) ) ) ), file(topalg_5,d5_topalg_5), []). fof(d7_borsuk_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_borsuk_2(D,A,B,C) => ! [E] : ( m1_borsuk_2(E,A,B,C) => ( r3_borsuk_2(A,B,C,D,E) <=> ? [F] : ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) & m2_relset_1(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) & v5_pre_topc(F,k6_borsuk_1(k5_topmetr,k5_topmetr),A) & ! [G] : ( m1_subset_1(G,u1_struct_0(k5_topmetr)) => ( k1_binop_1(F,G,0) = k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),D,G) & k1_binop_1(F,G,1) = k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),E,G) & ! [H] : ( m1_subset_1(H,u1_struct_0(k5_topmetr)) => ( k1_binop_1(F,0,H) = B & k1_binop_1(F,1,H) = C ) ) ) ) ) ) ) ) ) ) ) ), file(borsuk_2,d7_borsuk_2), []). fof(d8_toprealb,axiom,( k9_toprealb = k23_euclid(1,0) ), file(toprealb,d8_toprealb), []). fof(de_c4__topalg_5,axiom,( c4__topalg_5 = 0 ), file(topalg_5,c4__topalg_5), []). fof(de_c5__topalg_5,axiom,( c5__topalg_5 = 1 ), file(topalg_5,c5__topalg_5), []). fof(dt_c4__topalg_5,axiom,( m1_subset_1(c4__topalg_5,u1_struct_0(k5_topmetr)) ), file(topalg_5,c4__topalg_5), []). fof(dt_c5__topalg_5,axiom,( m1_subset_1(c5__topalg_5,u1_struct_0(k5_topmetr)) ), file(topalg_5,c5__topalg_5), []). fof(dt_g1_group_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) ) => ( v1_group_1(g1_group_1(A,B)) & l1_group_1(g1_group_1(A,B)) ) ) ), file(group_1,g1_group_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_toprealb,axiom, ( v1_funct_1(k12_toprealb) & v1_funct_2(k12_toprealb,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) & m2_relset_1(k12_toprealb,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) ), file(toprealb,k12_toprealb), []). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), []). fof(dt_k19_sin_cos,axiom,( $true ), file(sin_cos,k19_sin_cos), []). 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_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), []). fof(dt_k1_topalg_1,axiom,( $true ), file(topalg_1,k1_topalg_1), []). fof(dt_k1_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_funct_1(k1_topalg_5(A)) & v1_funct_2(k1_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) & m2_relset_1(k1_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) ) ) ), file(topalg_5,k1_topalg_5), []). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), []). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), []). fof(dt_k20_sin_cos,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k20_sin_cos(A),k1_numbers) ) ), file(sin_cos,k20_sin_cos), []). fof(dt_k22_borsuk_1,axiom,( l1_pre_topc(k22_borsuk_1) ), file(borsuk_1,k22_borsuk_1), []). fof(dt_k22_sin_cos,axiom,( $true ), file(sin_cos,k22_sin_cos), []). fof(dt_k23_euclid,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k23_euclid(A,B),u1_struct_0(k15_euclid(2))) ) ), file(euclid,k23_euclid), []). fof(dt_k23_sin_cos,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k23_sin_cos(A),k1_numbers) ) ), file(sin_cos,k23_sin_cos), []). fof(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), []). fof(dt_k2_topalg_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => m2_relset_1(k2_topalg_1(A,B),k1_topalg_1(A,B),k1_topalg_1(A,B)) ) ), file(topalg_1,k2_topalg_1), []). fof(dt_k2_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => m1_borsuk_2(k2_topalg_5(A),k3_topalg_2,k4_toprealb(0),k4_toprealb(A)) ) ), file(topalg_5,k2_topalg_5), []). 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_k31_sin_cos,axiom,( v1_xreal_0(k31_sin_cos) ), file(sin_cos,k31_sin_cos), []). fof(dt_k32_sin_cos,axiom,( m1_subset_1(k32_sin_cos,k1_numbers) ), file(sin_cos,k32_sin_cos), []). fof(dt_k3_gr_cy_1,axiom, ( ~ v3_struct_0(k3_gr_cy_1) & v1_group_1(k3_gr_cy_1) & v3_group_1(k3_gr_cy_1) & v4_group_1(k3_gr_cy_1) & l1_group_1(k3_gr_cy_1) ), file(gr_cy_1,k3_gr_cy_1), []). fof(dt_k3_topalg_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ( v1_group_1(k3_topalg_1(A,B)) & l1_group_1(k3_topalg_1(A,B)) ) ) ), file(topalg_1,k3_topalg_1), []). fof(dt_k3_topalg_2,axiom, ( v1_pre_topc(k3_topalg_2) & v3_topalg_2(k3_topalg_2) & m1_pre_topc(k3_topalg_2,k3_topmetr) ), file(topalg_2,k3_topalg_2), []). 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_borsuk_1,axiom,( ! [A,B,C,D,E] : ( ( ~ 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(C) & v2_pre_topc(C) & l1_pre_topc(C) & v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(D,A,B) & m1_relset_1(D,u1_struct_0(A),u1_struct_0(B)) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C)) & v5_pre_topc(E,B,C) & m1_relset_1(E,u1_struct_0(B),u1_struct_0(C)) ) => ( v1_funct_1(k4_borsuk_1(A,B,C,D,E)) & v1_funct_2(k4_borsuk_1(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C)) & v5_pre_topc(k4_borsuk_1(A,B,C,D,E),A,C) & m2_relset_1(k4_borsuk_1(A,B,C,D,E),u1_struct_0(A),u1_struct_0(C)) ) ) ), file(borsuk_1,k4_borsuk_1), []). 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_tarski,axiom,( $true ), file(tarski,k4_tarski), []). fof(dt_k4_topalg_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v3_topalg_2(A) & m1_pre_topc(A,k3_topalg_2) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ( v1_funct_1(k4_topalg_2(A,B,C,D,E)) & v1_funct_2(k4_topalg_2(A,B,C,D,E),u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) & m2_relset_1(k4_topalg_2(A,B,C,D,E),u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) ) ) ), file(topalg_2,k4_topalg_2), []). fof(dt_k4_toprealb,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k4_toprealb(A),u1_struct_0(k3_topalg_2)) ) ), file(toprealb,k4_toprealb), []). 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_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_topalg_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v3_topalg_2(A) & m1_pre_topc(A,k3_topalg_2) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => m1_borsuk_6(k5_topalg_2(A,B,C,D,E),A,B,C,D,E) ) ), file(topalg_2,k5_topalg_2), []). fof(dt_k5_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_funct_1(k5_topalg_5(A)) & v1_funct_2(k5_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(2))) & m2_relset_1(k5_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k8_toprealb(2))) ) ) ), file(topalg_5,k5_topalg_5), []). fof(dt_k5_topmetr,axiom, ( v1_pre_topc(k5_topmetr) & m1_pre_topc(k5_topmetr,k3_topmetr) ), file(topmetr,k5_topmetr), []). fof(dt_k6_borsuk_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & v2_pre_topc(B) & l1_pre_topc(B) ) => ( v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & l1_pre_topc(k6_borsuk_1(A,B)) ) ) ), file(borsuk_1,k6_borsuk_1), []). fof(dt_k6_eqrel_1,axiom,( ! [A,B,C] : ( ( v1_relat_2(B) & v3_relat_2(B) & v1_partfun1(B,A,A) & m1_relset_1(B,A,A) ) => m1_subset_1(k6_eqrel_1(A,B,C),k1_zfmisc_1(A)) ) ), file(eqrel_1,k6_eqrel_1), []). fof(dt_k6_topalg_5,axiom,( ! [A] : ( v1_int_1(A) => m1_borsuk_2(k6_topalg_5(A),k8_toprealb(2),k9_toprealb,k9_toprealb) ) ), file(topalg_5,k6_topalg_5), []). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), []). fof(dt_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => ( v1_funct_1(k7_funct_2(A,B,C,D,E)) & v1_funct_2(k7_funct_2(A,B,C,D,E),A,C) & m2_relset_1(k7_funct_2(A,B,C,D,E),A,C) ) ) ), file(funct_2,k7_funct_2), []). fof(dt_k7_topalg_5,axiom, ( v1_funct_1(k7_topalg_5) & v1_funct_2(k7_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(2),k9_toprealb))) & m2_relset_1(k7_topalg_5,u1_struct_0(k3_gr_cy_1),u1_struct_0(k3_topalg_1(k8_toprealb(2),k9_toprealb))) ), file(topalg_5,k7_topalg_5), []). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), []). fof(dt_k8_toprealb,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_pre_topc(k8_toprealb(A),k15_euclid(A)) ) ), file(toprealb,k8_toprealb), []). fof(dt_k9_toprealb,axiom,( m1_subset_1(k9_toprealb,u1_struct_0(k8_toprealb(2))) ), file(toprealb,k9_toprealb), []). fof(dt_l1_group_1,axiom,( ! [A] : ( l1_group_1(A) => l1_struct_0(A) ) ), file(group_1,l1_group_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_borsuk_2,axiom,( ! [A,B,C] : ( ( l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ! [D] : ( m1_borsuk_2(D,A,B,C) => ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) ) ) ), file(borsuk_2,m1_borsuk_2), []). fof(dt_m1_borsuk_6,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ! [F] : ( m1_borsuk_6(F,A,B,C,D,E) => ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) & m2_relset_1(F,u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)),u1_struct_0(A)) ) ) ) ), file(borsuk_6,m1_borsuk_6), []). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), []). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), []). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), []). fof(dt_u1_group_1,axiom,( ! [A] : ( l1_group_1(A) => ( v1_funct_1(u1_group_1(A)) & v1_funct_2(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) & m2_relset_1(u1_group_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ), file(group_1,u1_group_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_group_1,axiom,( ? [A] : l1_group_1(A) ), file(group_1,l1_group_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_borsuk_2,axiom,( ! [A,B,C] : ( ( l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) ) => ? [D] : m1_borsuk_2(D,A,B,C) ) ), file(borsuk_2,m1_borsuk_2), []). fof(existence_m1_borsuk_6,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ? [F] : m1_borsuk_6(F,A,B,C,D,E) ) ), file(borsuk_6,m1_borsuk_6), []). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), []). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), []). fof(fc10_borsuk_5,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & ~ v1_rat_1(k3_xcmplx_0(A,B)) ) ) ), file(borsuk_5,fc10_borsuk_5), []). 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_borsuk_5,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_xcmplx_0(k3_xcmplx_0(B,A)) & ~ v1_rat_1(k3_xcmplx_0(B,A)) ) ) ), file(borsuk_5,fc11_borsuk_5), []). 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_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_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_borsuk_5,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v1_connsp_1(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) & v1_borsuk_2(k22_borsuk_1) & v1_urysohn1(k22_borsuk_1) ), file(borsuk_5,fc18_borsuk_5), []). fof(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_borsuk_5,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & v1_connsp_1(k3_topmetr) & v2_t_0topsp(k3_topmetr) & v3_compts_1(k3_topmetr) & v1_borsuk_2(k3_topmetr) & v1_urysohn1(k3_topmetr) ), file(borsuk_5,fc1_borsuk_5), []). fof(fc1_funct_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_relat_1(B) & v1_funct_1(B) ) => ( v1_relat_1(k5_relat_1(A,B)) & v1_funct_1(k5_relat_1(A,B)) ) ) ), file(funct_1,fc1_funct_1), []). fof(fc1_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), []). fof(fc1_nat_1,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), []). fof(fc1_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), []). fof(fc1_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), []). fof(fc1_topalg_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ~ v1_xboole_0(k1_topalg_1(A,B)) ) ), file(topalg_1,fc1_topalg_1), []). fof(fc1_topalg_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) ) => ( v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v2_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v1_connsp_1(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ), file(topalg_3,fc1_topalg_3), []). fof(fc1_topalg_5,axiom, ( ~ v3_struct_0(k3_gr_cy_1) & v1_group_1(k3_gr_cy_1) & v3_group_1(k3_gr_cy_1) & v4_group_1(k3_gr_cy_1) & ~ v6_group_1(k3_gr_cy_1) ), file(topalg_5,fc1_topalg_5), []). fof(fc1_xboole_0,axiom,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), []). 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(fc1_yellow13,axiom,( ! [A] : ( ( v6_group_1(A) & l1_struct_0(A) ) => v1_finset_1(u1_struct_0(A)) ) ), file(yellow13,fc1_yellow13), []). fof(fc20_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_toprealb,axiom,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) ) => ( ~ v3_struct_0(k8_toprealb(A)) & v2_pre_topc(k8_toprealb(A)) & v1_monoid_0(k8_toprealb(A)) & v2_monoid_0(k8_toprealb(A)) ) ) ), file(toprealb,fc23_toprealb), []). fof(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_toprealb,axiom, ( ~ v3_struct_0(k8_toprealb(2)) & v2_pre_topc(k8_toprealb(2)) & v2_compts_1(k8_toprealb(2)) & v10_pscomp_1(k8_toprealb(2)) & v1_borsuk_2(k8_toprealb(2)) & v1_monoid_0(k8_toprealb(2)) & v2_monoid_0(k8_toprealb(2)) & v1_toprealb(k8_toprealb(2)) ), file(toprealb,fc24_toprealb), []). fof(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(fc28_toprealb,axiom, ( v1_relat_1(k12_toprealb) & v1_funct_1(k12_toprealb) & v1_funct_2(k12_toprealb,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) & v5_pre_topc(k12_toprealb,k3_topalg_2,k8_toprealb(2)) ), file(toprealb,fc28_toprealb), []). fof(fc29_toprealb,axiom, ( v1_relat_1(k12_toprealb) & v1_funct_1(k12_toprealb) & v1_funct_2(k12_toprealb,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) & v2_funct_2(k12_toprealb,u1_struct_0(k3_topalg_2),u1_struct_0(k8_toprealb(2))) & v5_pre_topc(k12_toprealb,k3_topalg_2,k8_toprealb(2)) ), file(toprealb,fc29_toprealb), []). fof(fc2_borsuk_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) ) ) ), file(borsuk_1,fc2_borsuk_1), []). fof(fc2_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_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_rcomp_3,axiom,( ! [A] : ( ( v1_borsuk_6(A) & l1_struct_0(A) ) => ( v1_membered(u1_struct_0(A)) & v2_membered(u1_struct_0(A)) ) ) ), file(rcomp_3,fc2_rcomp_3), []). fof(fc2_subset_1,axiom,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), []). fof(fc2_topalg_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ( v1_relat_1(k2_topalg_1(A,B)) & ~ v1_xboole_0(k2_topalg_1(A,B)) & v1_relat_2(k2_topalg_1(A,B)) & v3_relat_2(k2_topalg_1(A,B)) & v8_relat_2(k2_topalg_1(A,B)) & v1_partfun1(k2_topalg_1(A,B),k1_topalg_1(A,B),k1_topalg_1(A,B)) ) ) ), file(topalg_1,fc2_topalg_1), []). fof(fc2_topalg_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v3_topalg_2(A) & m1_pre_topc(A,k3_topalg_2) & m1_subset_1(B,u1_struct_0(A)) ) => ( ~ v3_struct_0(k3_topalg_1(A,B)) & v1_group_1(k3_topalg_1(A,B)) & v2_group_1(k3_topalg_1(A,B)) & v3_group_1(k3_topalg_1(A,B)) & v4_group_1(k3_topalg_1(A,B)) & v3_realset2(k3_topalg_1(A,B)) ) ) ), file(topalg_2,fc2_topalg_2), []). fof(fc2_topalg_3,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) & v2_pre_topc(B) & v1_connsp_1(B) & l1_pre_topc(B) ) => ( v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v1_connsp_1(k6_borsuk_1(A,B)) ) ) ), file(topalg_3,fc2_topalg_3), []). fof(fc2_topalg_5,axiom,( ! [A] : ( ( ~ v6_group_1(A) & l1_struct_0(A) ) => ~ v1_finset_1(u1_struct_0(A)) ) ), file(topalg_5,fc2_topalg_5), []). fof(fc2_yellow13,axiom,( ! [A] : ( ( v3_realset2(A) & l1_struct_0(A) ) => v1_realset1(u1_struct_0(A)) ) ), file(yellow13,fc2_yellow13), []). fof(fc3_int_1,axiom,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), []). 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_realset1,axiom,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) & v1_realset1(k1_tarski(A)) ) ), file(realset1,fc3_realset1), []). fof(fc3_sin_cos,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k19_sin_cos(A)) & v1_xreal_0(k19_sin_cos(A)) ) ) ), file(sin_cos,fc3_sin_cos), []). fof(fc3_subset_1,axiom,( ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ), file(subset_1,fc3_subset_1), []). fof(fc3_topalg_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ( ~ v3_struct_0(k3_topalg_1(A,B)) & v1_group_1(k3_topalg_1(A,B)) ) ) ), file(topalg_1,fc3_topalg_1), []). fof(fc3_topalg_2,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ( ~ v3_struct_0(k3_topalg_1(k5_topmetr,A)) & v1_group_1(k3_topalg_1(k5_topmetr,A)) & v2_group_1(k3_topalg_1(k5_topmetr,A)) & v3_group_1(k3_topalg_1(k5_topmetr,A)) & v4_group_1(k3_topalg_1(k5_topmetr,A)) & v3_realset2(k3_topalg_1(k5_topmetr,A)) ) ) ), file(topalg_2,fc3_topalg_2), []). 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_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_t_0topsp(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & v2_t_0topsp(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v2_t_0topsp(k6_borsuk_1(A,B)) ) ) ), file(borsuk_2,fc4_borsuk_2), []). fof(fc4_int_1,axiom,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), []). 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_sin_cos,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k22_sin_cos(A)) & v1_xreal_0(k22_sin_cos(A)) ) ) ), file(sin_cos,fc4_sin_cos), []). fof(fc4_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), []). fof(fc4_topalg_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ( ~ v3_struct_0(k3_topalg_1(A,B)) & v1_group_1(k3_topalg_1(A,B)) & v2_group_1(k3_topalg_1(A,B)) & v3_group_1(k3_topalg_1(A,B)) & v4_group_1(k3_topalg_1(A,B)) ) ) ), file(topalg_1,fc4_topalg_1), []). fof(fc4_topalg_5,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v1_connsp_2(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) ), file(topalg_5,fc4_topalg_5), []). fof(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_borsuk_1,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) ), file(borsuk_1,fc5_borsuk_1), []). fof(fc5_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_urysohn1(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & v1_urysohn1(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v1_urysohn1(k6_borsuk_1(A,B)) ) ) ), file(borsuk_2,fc5_borsuk_2), []). fof(fc5_borsuk_5,axiom,( ! [A] : ( ( v1_xreal_0(A) & ~ v1_rat_1(A) ) => ( ~ v1_xboole_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_xcmplx_0(k4_xcmplx_0(A)) & ~ v1_rat_1(k4_xcmplx_0(A)) ) ) ), file(borsuk_5,fc5_borsuk_5), []). fof(fc5_int_1,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), []). fof(fc5_membered,axiom, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), []). fof(fc5_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(fc5_yellow12,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ( ~ v3_struct_0(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ), file(yellow12,fc5_yellow12), []). fof(fc6_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v3_compts_1(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & v3_compts_1(B) & l1_pre_topc(B) ) => ( ~ v3_struct_0(k6_borsuk_1(A,B)) & v1_pre_topc(k6_borsuk_1(A,B)) & v2_pre_topc(k6_borsuk_1(A,B)) & v2_t_0topsp(k6_borsuk_1(A,B)) & v3_compts_1(k6_borsuk_1(A,B)) & v3_yellow_8(k6_borsuk_1(A,B)) ) ) ), file(borsuk_2,fc6_borsuk_2), []). fof(fc6_borsuk_5,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & ~ v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(borsuk_5,fc6_borsuk_5), []). fof(fc6_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc6_topalg_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => ( ~ v3_struct_0(k3_topalg_1(k15_euclid(A),B)) & v1_group_1(k3_topalg_1(k15_euclid(A),B)) & v2_group_1(k3_topalg_1(k15_euclid(A),B)) & v3_group_1(k3_topalg_1(k15_euclid(A),B)) & v4_group_1(k3_topalg_1(k15_euclid(A),B)) & v3_realset2(k3_topalg_1(k15_euclid(A),B)) ) ) ), file(topalg_1,fc6_topalg_1), []). fof(fc6_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_relat_1(k1_topalg_5(A)) & v1_funct_1(k1_topalg_5(A)) & v1_funct_2(k1_topalg_5(A),u1_struct_0(k5_topmetr),u1_struct_0(k3_topalg_2)) & v5_pre_topc(k1_topalg_5(A),k5_topmetr,k3_topalg_2) ) ) ), file(topalg_5,fc6_topalg_5), []). fof(fc6_yellow12,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) & v2_pre_topc(g1_pre_topc(u1_struct_0(A),u1_pre_topc(A))) ) ) ), file(yellow12,fc6_yellow12), []). fof(fc7_borsuk_2,axiom, ( ~ v3_struct_0(k22_borsuk_1) & v1_pre_topc(k22_borsuk_1) & v2_pre_topc(k22_borsuk_1) & v2_t_0topsp(k22_borsuk_1) & v2_compts_1(k22_borsuk_1) & v3_compts_1(k22_borsuk_1) & v3_yellow_8(k22_borsuk_1) ), file(borsuk_2,fc7_borsuk_2), []). fof(fc7_borsuk_5,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & ~ v1_rat_1(k2_xcmplx_0(B,A)) ) ) ), file(borsuk_5,fc7_borsuk_5), []). fof(fc7_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_1), []). 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_borsuk_5,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & ~ v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), file(borsuk_5,fc8_borsuk_5), []). fof(fc8_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), []). 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_borsuk_5,axiom,( ! [A,B] : ( ( v1_rat_1(A) & v1_xreal_0(B) & ~ v1_rat_1(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & ~ v1_rat_1(k6_xcmplx_0(B,A)) ) ) ), file(borsuk_5,fc9_borsuk_5), []). fof(fc9_int_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_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(free_g1_group_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),A) & m1_relset_1(B,k2_zfmisc_1(A,A),A) ) => ! [C,D] : ( g1_group_1(A,B) = g1_group_1(C,D) => ( A = C & B = D ) ) ) ), file(group_1,g1_group_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(l5_topalg_5,axiom,( u1_struct_0(k6_borsuk_1(k5_topmetr,k5_topmetr)) = k2_zfmisc_1(u1_struct_0(k5_topmetr),u1_struct_0(k5_topmetr)) ), file(topalg_5,l5_topalg_5), []). fof(rc1_borsuk_2,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_relset_1(B,u1_struct_0(A),u1_struct_0(A)) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A)) & v5_pre_topc(B,A,A) & v1_partfun1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ), file(borsuk_2,rc1_borsuk_2), []). fof(rc1_borsuk_5,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v1_connsp_1(A) ) ), file(borsuk_5,rc1_borsuk_5), []). fof(rc1_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), []). fof(rc1_int_1,axiom,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), []). fof(rc1_membered,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), []). fof(rc1_nat_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), []). fof(rc1_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), []). fof(rc1_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(realset1,rc1_realset1), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_topreal6,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v2_compts_1(A) & v6_group_1(A) & v10_pscomp_1(A) ) ), file(topreal6,rc1_topreal6), []). fof(rc1_toprealb,axiom,( ? [A] : ( m1_pre_topc(A,k15_euclid(2)) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v2_compts_1(A) & v10_pscomp_1(A) & v1_borsuk_2(A) & v1_monoid_0(A) & v2_monoid_0(A) & v1_toprealb(A) ) ), file(toprealb,rc1_toprealb), []). fof(rc1_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) ) ) ), file(tops_1,rc1_tops_1), []). fof(rc1_xboole_0,axiom,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_borsuk_1,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ? [C] : ( m1_relset_1(C,u1_struct_0(A),u1_struct_0(B)) & ~ v1_xboole_0(C) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(C,A,B) & v1_partfun1(C,u1_struct_0(A),u1_struct_0(B)) ) ) ), file(borsuk_1,rc2_borsuk_1), []). fof(rc2_borsuk_2,axiom,( ! [A,B] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) ) => ? [C] : ( m1_borsuk_2(C,A,B,B) & ~ v1_xboole_0(C) & v1_relat_1(C) & v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) & v5_pre_topc(C,k5_topmetr,A) & v1_partfun1(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) ) ), file(borsuk_2,rc2_borsuk_2), []). fof(rc2_borsuk_5,axiom,( ? [A] : ( v1_xreal_0(A) & v1_xcmplx_0(A) & ~ v1_rat_1(A) ) ), file(borsuk_5,rc2_borsuk_5), []). fof(rc2_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), []). fof(rc2_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), []). fof(rc2_pcomps_1,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v3_compts_1(A) ) ), file(pcomps_1,rc2_pcomps_1), []). fof(rc2_pre_topc,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), []). fof(rc2_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) ), file(realset1,rc2_realset1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ), file(tops_1,rc2_tops_1), []). fof(rc2_xboole_0,axiom,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), []). fof(rc2_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), []). fof(rc3_borsuk_2,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) ) ), file(borsuk_2,rc3_borsuk_2), []). fof(rc3_borsuk_5,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xreal_0(A) & v1_xcmplx_0(A) & v1_rat_1(A) ) ), file(borsuk_5,rc3_borsuk_5), []). fof(rc3_funct_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), []). fof(rc3_nat_1,axiom,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), []). fof(rc3_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), []). fof(rc3_struct_0,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), []). fof(rc3_topalg_2,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) & ~ v1_xboole_0(A) & v2_topalg_2(A) ) ), file(topalg_2,rc3_topalg_2), []). fof(rc3_tops_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ), file(tops_1,rc3_tops_1), []). fof(rc3_xreal_0,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), []). fof(rc4_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), []). fof(rc4_rcomp_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v2_connsp_1(B,A) ) ) ), file(rcomp_3,rc4_rcomp_3), []). fof(rc4_topalg_2,axiom,( ? [A] : ( m1_pre_topc(A,k3_topmetr) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v1_borsuk_6(A) & v3_topalg_2(A) ) ), file(topalg_2,rc4_topalg_2), []). fof(rc4_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) ) ) ), file(tops_1,rc4_tops_1), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), []). fof(rc5_rcomp_3,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v2_connsp_1(B,A) ) ) ), file(rcomp_3,rc5_rcomp_3), []). fof(rc5_struct_0,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), []). fof(rc5_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) & v3_tops_1(B,A) ) ) ), file(tops_1,rc5_tops_1), []). fof(rc6_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc6_pre_topc), []). fof(rc7_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc7_pre_topc), []). fof(redefinition_k20_sin_cos,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k20_sin_cos(A) = k19_sin_cos(A) ) ), file(sin_cos,k20_sin_cos), []). fof(redefinition_k23_sin_cos,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => k23_sin_cos(A) = k22_sin_cos(A) ) ), file(sin_cos,k23_sin_cos), []). fof(redefinition_k2_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => k2_topalg_5(A) = k1_topalg_5(A) ) ), file(topalg_5,k2_topalg_5), []). fof(redefinition_k32_sin_cos,axiom,( k32_sin_cos = k31_sin_cos ), file(sin_cos,k32_sin_cos), []). fof(redefinition_k3_topalg_2,axiom,( k3_topalg_2 = k3_topmetr ), file(topalg_2,k3_topalg_2), []). fof(redefinition_k4_borsuk_1,axiom,( ! [A,B,C,D,E] : ( ( ~ 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(C) & v2_pre_topc(C) & l1_pre_topc(C) & v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(D,A,B) & m1_relset_1(D,u1_struct_0(A),u1_struct_0(B)) & v1_funct_1(E) & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C)) & v5_pre_topc(E,B,C) & m1_relset_1(E,u1_struct_0(B),u1_struct_0(C)) ) => k4_borsuk_1(A,B,C,D,E) = k5_relat_1(D,E) ) ), file(borsuk_1,k4_borsuk_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_k5_numbers,axiom,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), []). fof(redefinition_k5_topalg_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v3_topalg_2(A) & m1_pre_topc(A,k3_topalg_2) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => k5_topalg_2(A,B,C,D,E) = k4_topalg_2(A,B,C,D,E) ) ), file(topalg_2,k5_topalg_2), []). fof(redefinition_k5_topmetr,axiom,( k5_topmetr = k22_borsuk_1 ), file(topmetr,k5_topmetr), []). fof(redefinition_k6_topalg_5,axiom,( ! [A] : ( v1_int_1(A) => k6_topalg_5(A) = k5_topalg_5(A) ) ), file(topalg_5,k6_topalg_5), []). fof(redefinition_k7_funct_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(B) & v1_funct_1(D) & v1_funct_2(D,A,B) & m1_relset_1(D,A,B) & v1_funct_1(E) & v1_funct_2(E,B,C) & m1_relset_1(E,B,C) ) => k7_funct_2(A,B,C,D,E) = k5_relat_1(D,E) ) ), file(funct_2,k7_funct_2), []). fof(redefinition_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), []). fof(redefinition_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), []). fof(redefinition_r4_borsuk_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ( r4_borsuk_2(A,B,C,D,E) <=> r3_borsuk_2(A,B,C,D,E) ) ) ), file(borsuk_2,r4_borsuk_2), []). fof(reflexivity_r1_tarski,axiom,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), []). fof(reflexivity_r4_borsuk_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => r4_borsuk_2(A,B,C,D,D) ) ), file(borsuk_2,r4_borsuk_2), []). fof(spc0_boole,axiom,( v1_xboole_0(0) ), file(boole,spc0_boole), []). fof(spc1_boole,axiom,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), []). fof(spc1_numerals,axiom, ( v2_xreal_0(1) & m1_subset_1(1,k5_numbers) ), file(numerals,spc1_numerals), []). fof(spc2_boole,axiom,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), []). fof(spc2_numerals,axiom, ( v2_xreal_0(2) & m1_subset_1(2,k5_numbers) ), file(numerals,spc2_numerals), []). fof(symmetry_r3_borsuk_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ( r3_borsuk_2(A,B,C,D,E) => r3_borsuk_2(A,B,C,E,D) ) ) ), file(borsuk_2,r3_borsuk_2), []). fof(symmetry_r4_borsuk_2,axiom,( ! [A,B,C,D,E] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_2(A) & l1_pre_topc(A) & m1_subset_1(B,u1_struct_0(A)) & m1_subset_1(C,u1_struct_0(A)) & m1_borsuk_2(D,A,B,C) & m1_borsuk_2(E,A,B,C) ) => ( r4_borsuk_2(A,B,C,D,E) => r4_borsuk_2(A,B,C,E,D) ) ) ), file(borsuk_2,r4_borsuk_2), []). fof(t10_complex2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_int_1(B) => k22_sin_cos(A) = k22_sin_cos(k2_xcmplx_0(k3_xcmplx_0(k4_real_1(2,k32_sin_cos),B),A)) ) ) ), file(complex2,t10_complex2), []). 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_subset,axiom,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), []). fof(t20_topalg_5,axiom,( ! [A] : ( v1_xreal_0(A) => k5_topalg_5(A) = k4_borsuk_1(k5_topmetr,k3_topalg_2,k8_toprealb(2),k2_topalg_5(A),k12_toprealb) ) ), file(topalg_5,t20_topalg_5), []). fof(t21_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ! [E] : ( ( v1_relat_1(E) & v1_funct_1(E) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | k1_funct_1(k5_relat_1(D,E),C) = k1_funct_1(E,k1_funct_1(D,C)) ) ) ) ) ), file(funct_2,t21_funct_2), []). fof(t2_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), []). fof(t2_subset,axiom,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), []). fof(t33_toprealb,axiom,( ! [A] : ( v1_int_1(A) => k1_funct_1(k12_toprealb,A) = k9_toprealb ) ), file(toprealb,t33_toprealb), []). fof(t34_sin_cos,axiom,( ! [A] : ( v1_xreal_0(A) => ( k23_sin_cos(0) = 1 & k20_sin_cos(0) = 0 & k22_sin_cos(k4_xcmplx_0(A)) = k22_sin_cos(A) & k19_sin_cos(k4_xcmplx_0(A)) = k4_xcmplx_0(k19_sin_cos(A)) ) ) ), file(sin_cos,t34_sin_cos), []). fof(t3_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), []). fof(t3_subset,axiom,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), []). fof(t3_topalg_3,axiom,( ! [A,B,C,D,E,F] : ( ( v1_funct_1(F) & v1_funct_2(F,k2_zfmisc_1(C,D),E) & m2_relset_1(F,k2_zfmisc_1(C,D),E) ) => ! [G] : ( ( v1_relat_1(G) & v1_funct_1(G) ) => ( ( r2_hidden(A,C) & r2_hidden(B,D) ) => ( E = k1_xboole_0 | k1_binop_1(k5_relat_1(F,G),A,B) = k1_funct_1(G,k1_binop_1(F,A,B)) ) ) ) ) ), file(topalg_3,t3_topalg_3), []). fof(t47_topalg_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_borsuk_2(C,A,B,B) => ! [D] : ( m1_borsuk_2(D,A,B,B) => ( k6_eqrel_1(k1_topalg_1(A,B),k2_topalg_1(A,B),C) = k6_eqrel_1(k1_topalg_1(A,B),k2_topalg_1(A,B),D) <=> r3_borsuk_2(A,B,B,C,D) ) ) ) ) ) ), file(topalg_1,t47_topalg_1), []). fof(t4_arithm,axiom,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), []). 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(t58_tops_2,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( ( ~ v3_struct_0(C) & l1_pre_topc(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(C),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) ) => ( ( v5_pre_topc(D,A,C) & v5_pre_topc(E,C,B) ) => v5_pre_topc(k7_funct_2(u1_struct_0(A),u1_struct_0(C),u1_struct_0(B),D,E),A,B) ) ) ) ) ) ) ), file(tops_2,t58_tops_2), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t7_boole,axiom,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), []). fof(t8_boole,axiom,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), []). fof(t9_complex2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_int_1(B) => k19_sin_cos(A) = k19_sin_cos(k2_xcmplx_0(k3_xcmplx_0(k4_real_1(2,k32_sin_cos),B),A)) ) ) ), file(complex2,t9_complex2), []).