% Mizar problem: t57_rcomp_3,rcomp_3,2009,19 fof(t57_rcomp_3,conjecture,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ! [C] : ( ( v3_pre_topc(C,k4_topmetr(A,B)) & v2_connsp_1(C,k4_topmetr(A,B)) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))) ) => ~ ( ~ v1_xboole_0(C) & C != k1_rcomp_1(A,B) & ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(D,A) & r1_xreal_0(D,B) & C = k1_rcomp_2(A,D) ) ) & ! [D] : ( v1_xreal_0(D) => ~ ( r1_xreal_0(A,D) & ~ r1_xreal_0(B,D) & C = k2_rcomp_2(D,B) ) ) & ! [D] : ( v1_xreal_0(D) => ! [E] : ( v1_xreal_0(E) => ~ ( r1_xreal_0(A,D) & ~ r1_xreal_0(E,D) & r1_xreal_0(E,B) & C = k2_rcomp_1(D,E) ) ) ) ) ) ) ) ) ), inference(mizar_bg_added,[status(thm)],[abstractness_v1_pre_topc,antisymmetry_r2_hidden,cc10_membered,cc10_yellow13,cc11_membered,cc11_topreal6,cc11_yellow13,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_borsuk_6,cc1_finseq_1,cc1_membered,cc1_pre_topc,cc1_realset1,cc1_sprect_1,cc1_topreal6,cc1_tops_1,cc1_xreal_0,cc1_yellow13,cc20_membered,cc2_borsuk_6,cc2_membered,cc2_pscomp_1,cc2_sprect_1,cc2_topreal6,cc2_tops_1,cc2_xreal_0,cc2_yellow13,cc3_borsuk_5,cc3_int_1,cc3_membered,cc3_nat_1,cc3_sprect_1,cc3_topalg_2,cc3_tops_1,cc3_xreal_0,cc3_yellow13,cc4_int_1,cc4_membered,cc4_sprect_1,cc4_topalg_2,cc4_tops_1,cc4_xreal_0,cc4_yellow13,cc5_tops_1,cc5_xreal_0,cc5_yellow13,cc6_membered,cc6_tops_1,cc6_xreal_0,cc6_yellow13,cc7_xreal_0,cc7_yellow13,cc8_xreal_0,cc8_yellow13,cc9_yellow13,connectedness_r1_xreal_0,d2_pre_topc,d3_pre_topc,d5_real_1,dt_g1_pre_topc,dt_k12_prob_1,dt_k1_numbers,dt_k1_pre_topc,dt_k1_rcomp_1,dt_k1_rcomp_2,dt_k1_tops_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_limfunc1,dt_k2_pre_topc,dt_k2_rcomp_1,dt_k2_rcomp_2,dt_k2_subset_1,dt_k3_limfunc1,dt_k3_topmetr,dt_k4_limfunc1,dt_k4_topmetr,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_pre_topc,dt_u1_struct_0,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,fc10_rcomp_3,fc10_tops_1,fc11_rcomp_3,fc12_rcomp_3,fc13_rcomp_3,fc14_borsuk_5,fc14_rcomp_3,fc15_borsuk_5,fc16_borsuk_5,fc17_borsuk_5,fc17_rcomp_3,fc1_borsuk_5,fc1_fcont_3,fc1_pre_topc,fc1_pscomp_1,fc1_rcomp_1,fc1_rcomp_3,fc1_struct_0,fc1_subset_1,fc1_tops_1,fc1_yellow13,fc2_finseq_1,fc2_membered,fc2_pre_topc,fc2_rcomp_1,fc2_rcomp_3,fc2_yellow13,fc3_fcont_3,fc3_rcomp_3,fc3_yellow13,fc4_fcont_3,fc4_rcomp_3,fc5_fcont_3,fc5_pre_topc,fc5_rcomp_3,fc5_tops_1,fc6_fcont_3,fc6_membered,fc6_rcomp_3,fc6_tops_1,fc7_borsuk_6,fc7_rcomp_3,fc7_topreal6,fc7_tops_1,fc8_borsuk_6,fc8_rcomp_3,fc8_topreal6,fc8_tops_1,fc9_rcomp_3,fc9_sprect_1,fc9_tops_1,free_g1_pre_topc,l1_rcomp_3,rc10_pscomp_1,rc1_borsuk_5,rc1_borsuk_6,rc1_finseq_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_pscomp_1,rc1_rcomp_1,rc1_rcomp_3,rc1_realset1,rc1_sprect_1,rc1_subset_1,rc1_topreal6,rc1_tops_1,rc1_xreal_0,rc2_borsuk_5,rc2_borsuk_6,rc2_int_1,rc2_pre_topc,rc2_pscomp_1,rc2_rcomp_3,rc2_realset1,rc2_subset_1,rc2_tops_1,rc2_waybel_7,rc2_xreal_0,rc3_borsuk_5,rc3_borsuk_6,rc3_finseq_1,rc3_pre_topc,rc3_pscomp_1,rc3_rcomp_3,rc3_sprect_1,rc3_struct_0,rc3_topalg_2,rc3_tops_1,rc3_waybel_7,rc3_xreal_0,rc4_pre_topc,rc4_pscomp_1,rc4_rcomp_3,rc4_tops_1,rc4_xreal_0,rc5_pre_topc,rc5_pscomp_1,rc5_rcomp_3,rc5_struct_0,rc5_tops_1,rc6_finseq_1,rc6_pre_topc,rc7_finseq_1,rc7_pre_topc,rc8_finseq_1,rc9_pscomp_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,t12_rcomp_3,t14_borsuk_4,t14_connsp_1,t14_rcomp_3,t15_borsuk_4,t15_rcomp_1,t1_real,t1_subset,t1_xboole_1,t23_treal_1,t25_topmetr,t2_real,t2_subset,t3_rcomp_1,t3_rcomp_2,t3_real,t3_subset,t41_rcomp_3,t44_topreala,t47_rcomp_1,t48_rcomp_1,t4_rcomp_1,t4_rcomp_2,t4_real,t4_subset,t54_rcomp_3,t55_rcomp_3,t55_tops_1,t56_rcomp_3,t5_real,t5_subset,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real]), [file(rcomp_3,t57_rcomp_3)]). 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_6,axiom,( ! [A] : ( ( v1_borsuk_6(A) & l1_struct_0(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(borsuk_6,cc1_borsuk_6), []). fof(cc1_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), []). fof(cc1_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_realset1,axiom,( ! [A] : ( ~ v1_realset1(A) => ~ v1_xboole_0(A) ) ), file(realset1,cc1_realset1), []). fof(cc1_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & ~ v5_seqm_3(A) ) => ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & ~ v1_realset1(A) ) ) ), file(sprect_1,cc1_sprect_1), []). fof(cc1_topreal6,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_connsp_1(B,A) ) ) ) ), file(topreal6,cc1_topreal6), []). fof(cc1_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => ( v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ) ) ), file(tops_1,cc1_tops_1), []). fof(cc1_xreal_0,axiom,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), []). fof(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_6,axiom,( ! [A] : ( ( v1_borsuk_6(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v1_borsuk_6(B) ) ) ), file(borsuk_6,cc2_borsuk_6), []). fof(cc2_membered,axiom,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), []). fof(cc2_pscomp_1,axiom,( ! [A] : ( l1_pre_topc(A) => ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) ) => ( ~ v3_struct_0(A) & v2_pre_topc(A) & v10_pscomp_1(A) ) ) ) ), file(pscomp_1,cc2_pscomp_1), []). fof(cc2_sprect_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_realset1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v5_seqm_3(A) ) ) ), file(sprect_1,cc2_sprect_1), []). fof(cc2_topreal6,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_finset_1(B) => v6_compts_1(B,A) ) ) ) ), file(topreal6,cc2_topreal6), []). fof(cc2_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v2_tops_1(B,A) ) ) ) ), file(tops_1,cc2_tops_1), []). fof(cc2_xreal_0,axiom,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), []). fof(cc2_yellow13,axiom,( ! [A] : ( l1_struct_0(A) => ( v3_realset2(A) => v6_group_1(A) ) ) ), file(yellow13,cc2_yellow13), []). 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_sprect_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v3_seq_4(A) => ( v1_membered(A) & v2_membered(A) & v1_seq_4(A) & v2_seq_4(A) ) ) ) ), file(sprect_1,cc3_sprect_1), []). fof(cc3_topalg_2,axiom,( ! [A] : ( m1_pre_topc(A,k3_topmetr) => ( v2_pre_topc(A) & v1_borsuk_6(A) ) ) ), file(topalg_2,cc3_topalg_2), []). fof(cc3_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_xboole_0(B) => v3_tops_1(B,A) ) ) ) ), file(tops_1,cc3_tops_1), []). fof(cc3_xreal_0,axiom,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), []). fof(cc3_yellow13,axiom,( ! [A] : ( l1_struct_0(A) => ( ~ v3_realset2(A) => ~ v3_struct_0(A) ) ) ), file(yellow13,cc3_yellow13), []). fof(cc4_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_sprect_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( ( v1_seq_4(A) & v2_seq_4(A) ) => ( v1_membered(A) & v2_membered(A) & v3_seq_4(A) ) ) ) ), file(sprect_1,cc4_sprect_1), []). 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_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_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_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_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_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_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(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(d2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => k1_pre_topc(A) = k1_xboole_0 ) ), file(pre_topc,d2_pre_topc), []). fof(d3_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,d3_pre_topc), []). fof(d5_real_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), []). fof(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_prob_1,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k12_prob_1(A),k1_zfmisc_1(k1_numbers)) ) ), file(prob_1,k12_prob_1), []). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), []). fof(dt_k1_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k1_pre_topc), []). fof(dt_k1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k1_rcomp_1), []). fof(dt_k1_rcomp_2,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_2,k1_rcomp_2), []). fof(dt_k1_tops_1,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => m1_subset_1(k1_tops_1(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ), file(tops_1,k1_tops_1), []). 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_limfunc1,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k2_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ), file(limfunc1,k2_limfunc1), []). fof(dt_k2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k2_pre_topc), []). fof(dt_k2_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_1,k2_rcomp_1), []). fof(dt_k2_rcomp_2,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k2_rcomp_2(A,B),k1_zfmisc_1(k1_numbers)) ) ), file(rcomp_2,k2_rcomp_2), []). fof(dt_k2_subset_1,axiom,( ! [A] : m1_subset_1(k2_subset_1(A),k1_zfmisc_1(A)) ), file(subset_1,k2_subset_1), []). fof(dt_k3_limfunc1,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k3_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ), file(limfunc1,k3_limfunc1), []). fof(dt_k3_topmetr,axiom, ( v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & l1_pre_topc(k3_topmetr) ), file(topmetr,k3_topmetr), []). fof(dt_k4_limfunc1,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k4_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ), file(limfunc1,k4_limfunc1), []). 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_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_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), []). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), []). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), []). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), []). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), []). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), []). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), []). fof(fc10_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k2_limfunc1(A)) & v1_membered(k2_limfunc1(A)) & v2_membered(k2_limfunc1(A)) & v1_seq_4(k2_limfunc1(A)) & ~ v2_seq_4(k2_limfunc1(A)) & v2_rcomp_1(k2_limfunc1(A)) & v1_jct_misc(k2_limfunc1(A)) ) ) ), file(rcomp_3,fc10_rcomp_3), []). fof(fc10_tops_1,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & v2_tops_1(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_xboole_0(k1_tops_1(A,B)) & v1_membered(k1_tops_1(A,B)) & v2_membered(k1_tops_1(A,B)) & v3_membered(k1_tops_1(A,B)) & v4_membered(k1_tops_1(A,B)) & v5_membered(k1_tops_1(A,B)) & v2_tops_1(k1_tops_1(A,B),A) ) ) ), file(tops_1,fc10_tops_1), []). fof(fc11_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k12_prob_1(A)) & v1_membered(k12_prob_1(A)) & v2_membered(k12_prob_1(A)) & v1_seq_4(k12_prob_1(A)) & ~ v2_seq_4(k12_prob_1(A)) & v3_rcomp_1(k12_prob_1(A)) & v1_jct_misc(k12_prob_1(A)) ) ) ), file(rcomp_3,fc11_rcomp_3), []). fof(fc12_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k3_limfunc1(A)) & v1_membered(k3_limfunc1(A)) & v2_membered(k3_limfunc1(A)) & ~ v1_seq_4(k3_limfunc1(A)) & v2_seq_4(k3_limfunc1(A)) & v2_rcomp_1(k3_limfunc1(A)) & v1_jct_misc(k3_limfunc1(A)) ) ) ), file(rcomp_3,fc12_rcomp_3), []). fof(fc13_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k4_limfunc1(A)) & v1_membered(k4_limfunc1(A)) & v2_membered(k4_limfunc1(A)) & ~ v1_seq_4(k4_limfunc1(A)) & v2_seq_4(k4_limfunc1(A)) & v3_rcomp_1(k4_limfunc1(A)) & v1_jct_misc(k4_limfunc1(A)) ) ) ), file(rcomp_3,fc13_rcomp_3), []). fof(fc14_borsuk_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k4_limfunc1(A)) & v1_membered(k4_limfunc1(A)) & v2_membered(k4_limfunc1(A)) ) ) ), file(borsuk_5,fc14_borsuk_5), []). fof(fc14_rcomp_3,axiom, ( ~ v1_xboole_0(k2_subset_1(k1_numbers)) & v1_membered(k2_subset_1(k1_numbers)) & v2_membered(k2_subset_1(k1_numbers)) & ~ v1_seq_4(k2_subset_1(k1_numbers)) & ~ v2_seq_4(k2_subset_1(k1_numbers)) & v2_rcomp_1(k2_subset_1(k1_numbers)) & v3_rcomp_1(k2_subset_1(k1_numbers)) & v1_jct_misc(k2_subset_1(k1_numbers)) ), file(rcomp_3,fc14_rcomp_3), []). fof(fc15_borsuk_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k2_limfunc1(A)) & v1_membered(k2_limfunc1(A)) & v2_membered(k2_limfunc1(A)) ) ) ), file(borsuk_5,fc15_borsuk_5), []). fof(fc16_borsuk_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k12_prob_1(A)) & v1_membered(k12_prob_1(A)) & v2_membered(k12_prob_1(A)) ) ) ), file(borsuk_5,fc16_borsuk_5), []). fof(fc17_borsuk_5,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_xboole_0(k3_limfunc1(A)) & v1_membered(k3_limfunc1(A)) & v2_membered(k3_limfunc1(A)) ) ) ), file(borsuk_5,fc17_borsuk_5), []). fof(fc17_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( ~ v3_struct_0(k4_topmetr(A,A)) & v1_pre_topc(k4_topmetr(A,A)) & v2_pre_topc(k4_topmetr(A,A)) & v2_compts_1(k4_topmetr(A,A)) & v6_group_1(k4_topmetr(A,A)) & v3_realset2(k4_topmetr(A,A)) & v10_pscomp_1(k4_topmetr(A,A)) & v1_borsuk_6(k4_topmetr(A,A)) ) ) ), file(rcomp_3,fc17_rcomp_3), []). fof(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_fcont_3,axiom, ( v1_membered(k2_subset_1(k1_numbers)) & v2_membered(k2_subset_1(k1_numbers)) & v2_rcomp_1(k2_subset_1(k1_numbers)) & v3_rcomp_1(k2_subset_1(k1_numbers)) ), file(fcont_3,fc1_fcont_3), []). fof(fc1_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => ( v1_xboole_0(k1_pre_topc(A)) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(pre_topc,fc1_pre_topc), []). fof(fc1_pscomp_1,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) & ~ v1_setfam_1(k1_numbers) ), file(pscomp_1,fc1_pscomp_1), []). fof(fc1_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_1(A,B)) & v2_membered(k2_rcomp_1(A,B)) & v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ), file(rcomp_1,fc1_rcomp_1), []). fof(fc1_rcomp_3,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ~ v1_xboole_0(k2_subset_1(A)) ) ), file(rcomp_3,fc1_rcomp_3), []). 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_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_xboole_0(k1_pre_topc(A)) & v4_pre_topc(k1_pre_topc(A),A) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(tops_1,fc1_tops_1), []). 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(fc2_finseq_1,axiom, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), []). fof(fc2_membered,axiom, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), []). fof(fc2_pre_topc,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(k2_pre_topc(A)) ) ), file(pre_topc,fc2_pre_topc), []). fof(fc2_rcomp_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ), file(rcomp_1,fc2_rcomp_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_yellow13,axiom,( ! [A] : ( ( v3_realset2(A) & l1_struct_0(A) ) => v1_realset1(u1_struct_0(A)) ) ), file(yellow13,fc2_yellow13), []). fof(fc3_fcont_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k4_limfunc1(A)) & v2_membered(k4_limfunc1(A)) & v3_rcomp_1(k4_limfunc1(A)) ) ) ), file(fcont_3,fc3_fcont_3), []). fof(fc3_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_2(A,B)) & v2_membered(k1_rcomp_2(A,B)) & v1_seq_4(k1_rcomp_2(A,B)) & v2_seq_4(k1_rcomp_2(A,B)) & v3_seq_4(k1_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc3_rcomp_3), []). fof(fc3_yellow13,axiom,( ! [A] : ( ( v2_compts_1(A) & l1_pre_topc(A) ) => ( v1_tops_1(k2_pre_topc(A),A) & v6_compts_1(k2_pre_topc(A),A) ) ) ), file(yellow13,fc3_yellow13), []). fof(fc4_fcont_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k12_prob_1(A)) & v2_membered(k12_prob_1(A)) & v3_rcomp_1(k12_prob_1(A)) ) ) ), file(fcont_3,fc4_fcont_3), []). fof(fc4_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_2(A,B)) & v2_membered(k2_rcomp_2(A,B)) & v1_seq_4(k2_rcomp_2(A,B)) & v2_seq_4(k2_rcomp_2(A,B)) & v3_seq_4(k2_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc4_rcomp_3), []). fof(fc5_fcont_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k3_limfunc1(A)) & v2_membered(k3_limfunc1(A)) & v2_rcomp_1(k3_limfunc1(A)) ) ) ), file(fcont_3,fc5_fcont_3), []). fof(fc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v4_pre_topc(k2_pre_topc(A),A) ) ), file(pre_topc,fc5_pre_topc), []). fof(fc5_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_1(A,B)) & v2_membered(k2_rcomp_1(A,B)) & v1_seq_4(k2_rcomp_1(A,B)) & v2_seq_4(k2_rcomp_1(A,B)) & v3_seq_4(k2_rcomp_1(A,B)) & v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ), file(rcomp_3,fc5_rcomp_3), []). fof(fc5_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_xboole_0(k1_tops_1(A,k1_pre_topc(A))) & v1_membered(k1_tops_1(A,k1_pre_topc(A))) & v2_membered(k1_tops_1(A,k1_pre_topc(A))) & v3_membered(k1_tops_1(A,k1_pre_topc(A))) & v4_membered(k1_tops_1(A,k1_pre_topc(A))) & v5_membered(k1_tops_1(A,k1_pre_topc(A))) ) ) ), file(tops_1,fc5_tops_1), []). fof(fc6_fcont_3,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k2_limfunc1(A)) & v2_membered(k2_limfunc1(A)) & v2_rcomp_1(k2_limfunc1(A)) ) ) ), file(fcont_3,fc6_fcont_3), []). fof(fc6_membered,axiom, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), []). fof(fc6_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v1_seq_4(k1_rcomp_1(A,B)) & v2_seq_4(k1_rcomp_1(A,B)) & v3_seq_4(k1_rcomp_1(A,B)) & v1_rcomp_1(k1_rcomp_1(A,B)) & v2_rcomp_1(k1_rcomp_1(A,B)) & v1_jct_misc(k1_rcomp_1(A,B)) ) ) ), file(rcomp_3,fc6_rcomp_3), []). fof(fc6_tops_1,axiom,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => v3_pre_topc(k1_tops_1(A,B),A) ) ), file(tops_1,fc6_tops_1), []). fof(fc7_borsuk_6,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & v1_connsp_1(k3_topmetr) & v3_compts_1(k3_topmetr) & v1_urysohn1(k3_topmetr) & v1_borsuk_2(k3_topmetr) & v2_t_0topsp(k3_topmetr) & v1_borsuk_6(k3_topmetr) ), file(borsuk_6,fc7_borsuk_6), []). fof(fc7_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_2(A,B)) & v2_membered(k1_rcomp_2(A,B)) & v1_seq_4(k1_rcomp_2(A,B)) & v2_seq_4(k1_rcomp_2(A,B)) & v3_seq_4(k1_rcomp_2(A,B)) & v1_jct_misc(k1_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc7_rcomp_3), []). fof(fc7_topreal6,axiom, ( ~ v3_struct_0(k3_topmetr) & v1_pre_topc(k3_topmetr) & v2_pre_topc(k3_topmetr) & v3_compts_1(k3_topmetr) ), file(topreal6,fc7_topreal6), []). fof(fc7_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_xboole_0(k1_pre_topc(A)) & v3_pre_topc(k1_pre_topc(A),A) & v4_pre_topc(k1_pre_topc(A),A) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(tops_1,fc7_tops_1), []). fof(fc8_borsuk_6,axiom,( ! [A] : ( l1_pre_topc(A) => ( v1_funct_1(k1_pre_topc(A)) & v1_xboole_0(k1_pre_topc(A)) & v6_compts_1(k1_pre_topc(A),A) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(borsuk_6,fc8_borsuk_6), []). fof(fc8_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_2(A,B)) & v2_membered(k2_rcomp_2(A,B)) & v1_seq_4(k2_rcomp_2(A,B)) & v2_seq_4(k2_rcomp_2(A,B)) & v3_seq_4(k2_rcomp_2(A,B)) & v1_jct_misc(k2_rcomp_2(A,B)) ) ) ), file(rcomp_3,fc8_rcomp_3), []). fof(fc8_topreal6,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v1_seq_4(k1_rcomp_1(A,B)) & v2_seq_4(k1_rcomp_1(A,B)) & v3_seq_4(k1_rcomp_1(A,B)) & v1_rcomp_1(k1_rcomp_1(A,B)) & v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ), file(topreal6,fc8_topreal6), []). fof(fc8_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v3_pre_topc(k2_pre_topc(A),A) & v4_pre_topc(k2_pre_topc(A),A) ) ) ), file(tops_1,fc8_tops_1), []). fof(fc9_rcomp_3,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_rcomp_1(A,B)) & v2_membered(k2_rcomp_1(A,B)) & v1_seq_4(k2_rcomp_1(A,B)) & v2_seq_4(k2_rcomp_1(A,B)) & v3_seq_4(k2_rcomp_1(A,B)) & v3_rcomp_1(k2_rcomp_1(A,B)) & v1_jct_misc(k2_rcomp_1(A,B)) ) ) ), file(rcomp_3,fc9_rcomp_3), []). fof(fc9_sprect_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k1_rcomp_1(A,B)) & v2_membered(k1_rcomp_1(A,B)) & v3_seq_4(k1_rcomp_1(A,B)) ) ) ), file(sprect_1,fc9_sprect_1), []). fof(fc9_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => v1_tops_1(k2_pre_topc(A),A) ) ), file(tops_1,fc9_tops_1), []). fof(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(l1_rcomp_3,axiom,( k1_numbers = k2_subset_1(k1_numbers) ), file(rcomp_3,l1_rcomp_3), []). fof(rc10_pscomp_1,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v2_compts_1(A) & v10_pscomp_1(A) ) ), file(pscomp_1,rc10_pscomp_1), []). 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_borsuk_6,axiom,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) & v1_borsuk_6(A) ) ), file(borsuk_6,rc1_borsuk_6), []). fof(rc1_finseq_1,axiom,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), []). fof(rc1_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_pscomp_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_setfam_1(A) ) ), file(pscomp_1,rc1_pscomp_1), []). fof(rc1_rcomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & v1_membered(A) & v2_membered(A) & v3_rcomp_1(A) ) ), file(rcomp_1,rc1_rcomp_1), []). fof(rc1_rcomp_3,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) & v1_membered(A) & v2_membered(A) & v1_seq_4(A) & v2_seq_4(A) ) ), file(rcomp_3,rc1_rcomp_3), []). fof(rc1_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_realset1(A) ) ), file(realset1,rc1_realset1), []). fof(rc1_sprect_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v5_seqm_3(A) ) ), file(sprect_1,rc1_sprect_1), []). fof(rc1_subset_1,axiom,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), []). fof(rc1_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_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) ) ) ), file(tops_1,rc1_tops_1), []). fof(rc1_xreal_0,axiom,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), []). fof(rc2_borsuk_5,axiom,( ? [A] : ( v1_xreal_0(A) & v1_xcmplx_0(A) & ~ v1_rat_1(A) ) ), file(borsuk_5,rc2_borsuk_5), []). fof(rc2_borsuk_6,axiom,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v2_pre_topc(A) & v1_borsuk_6(A) ) ), file(borsuk_6,rc2_borsuk_6), []). fof(rc2_int_1,axiom,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_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_pscomp_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_setfam_1(A) ) ), file(pscomp_1,rc2_pscomp_1), []). fof(rc2_rcomp_3,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v1_seq_4(A) & v2_seq_4(A) & v3_seq_4(A) & v3_rcomp_1(A) & v1_jct_misc(A) ) ), file(rcomp_3,rc2_rcomp_3), []). fof(rc2_realset1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & ~ v1_realset1(A) ) ), file(realset1,rc2_realset1), []). fof(rc2_subset_1,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), []). fof(rc2_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) ) ) ), file(tops_1,rc2_tops_1), []). fof(rc2_waybel_7,axiom,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ), file(waybel_7,rc2_waybel_7), []). 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_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_borsuk_6,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_funct_1(B) & v1_xboole_0(B) & v6_compts_1(B,A) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ), file(borsuk_6,rc3_borsuk_6), []). fof(rc3_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), []). fof(rc3_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_pscomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v1_setfam_1(A) ) ), file(pscomp_1,rc3_pscomp_1), []). fof(rc3_rcomp_3,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & ~ v3_seq_4(A) & v2_rcomp_1(A) & v3_rcomp_1(A) & v1_jct_misc(A) ) ), file(rcomp_3,rc3_rcomp_3), []). fof(rc3_sprect_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v1_seq_4(A) & v2_seq_4(A) ) ), file(sprect_1,rc3_sprect_1), []). 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_waybel_7,axiom,( ! [A] : ( l1_struct_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(waybel_7,rc3_waybel_7), []). 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_pscomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & ~ v1_setfam_1(A) ) ), file(pscomp_1,rc4_pscomp_1), []). fof(rc4_rcomp_3,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(B,A) & v4_pre_topc(B,A) & v2_connsp_1(B,A) ) ) ), file(rcomp_3,rc4_rcomp_3), []). fof(rc4_tops_1,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v1_xboole_0(B) & v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) & v2_tops_1(B,A) ) ) ), file(tops_1,rc4_tops_1), []). fof(rc4_xreal_0,axiom,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), []). fof(rc5_pre_topc,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), []). fof(rc5_pscomp_1,axiom,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_seq_4(A) & v2_rcomp_1(A) ) ), file(pscomp_1,rc5_pscomp_1), []). 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_finseq_1,axiom,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), []). fof(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_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), []). fof(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(rc8_finseq_1,axiom,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), []). fof(rc9_pscomp_1,axiom,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) & v6_compts_1(B,A) ) ) ), file(pscomp_1,rc9_pscomp_1), []). fof(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(t12_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_tarski(k1_rcomp_2(A,B),k1_rcomp_1(C,D)) => ( r1_xreal_0(B,A) | ( r1_xreal_0(C,A) & r1_xreal_0(B,D) ) ) ) ) ) ) ) ), file(rcomp_3,t12_rcomp_3), []). fof(t14_borsuk_4,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_tarski(k1_rcomp_1(A,B),k1_rcomp_1(C,D)) ) => ( r1_xreal_0(C,A) & r1_xreal_0(B,D) ) ) ) ) ) ) ), file(borsuk_4,t14_borsuk_4), []). fof(t14_connsp_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_connsp_1(A) <=> ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v3_pre_topc(B,A) & v4_pre_topc(B,A) & B != k1_pre_topc(A) & B != k2_pre_topc(A) ) ) ) ) ), file(connsp_1,t14_connsp_1), []). fof(t14_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_tarski(k2_rcomp_2(A,B),k1_rcomp_1(C,D)) => ( r1_xreal_0(B,A) | ( r1_xreal_0(C,A) & r1_xreal_0(B,D) ) ) ) ) ) ) ) ), file(rcomp_3,t14_rcomp_3), []). fof(t15_borsuk_4,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(C,D)) => ( r1_xreal_0(B,A) | ( r1_xreal_0(C,A) & r1_xreal_0(B,D) ) ) ) ) ) ) ) ), file(borsuk_4,t15_borsuk_4), []). fof(t15_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ ( ~ r1_xreal_0(B,A) & k2_rcomp_1(A,B) = k1_xboole_0 ) & ( r1_xreal_0(A,B) => ( r2_hidden(A,k1_rcomp_1(A,B)) & r2_hidden(B,k1_rcomp_1(A,B)) ) ) & r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(A,B)) ) ) ) ), file(rcomp_1,t15_rcomp_1), []). 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(t23_treal_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => v1_connsp_1(k4_topmetr(A,B)) ) ) ) ), file(treal_1,t23_treal_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(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(t3_rcomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_numbers)) => ( ( r1_tarski(A,B) & v2_seq_4(B) ) => v2_seq_4(A) ) ) ) ), file(rcomp_1,t3_rcomp_1), []). fof(t3_rcomp_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r2_hidden(A,k1_rcomp_2(B,C)) <=> ( r1_xreal_0(B,A) & ~ r1_xreal_0(C,A) ) ) ) ) ) ), file(rcomp_2,t3_rcomp_2), []). 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(t41_rcomp_3,axiom,( ! [A] : ( ( v1_jct_misc(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ~ ( ~ v1_xboole_0(A) & A != k1_numbers & ! [B] : ( v1_xreal_0(B) => A != k2_limfunc1(B) ) & ! [B] : ( v1_xreal_0(B) => A != k12_prob_1(B) ) & ! [B] : ( v1_xreal_0(B) => A != k3_limfunc1(B) ) & ! [B] : ( v1_xreal_0(B) => A != k4_limfunc1(B) ) & ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( r1_xreal_0(B,C) & A = k1_rcomp_1(B,C) ) ) ) & ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,B) & A = k1_rcomp_2(B,C) ) ) ) & ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,B) & A = k2_rcomp_2(B,C) ) ) ) & ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,B) & A = k2_rcomp_1(B,C) ) ) ) ) ) ), file(rcomp_3,t41_rcomp_3), []). fof(t44_topreala,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => ( v4_pre_topc(k1_rcomp_1(B,C),k4_topmetr(A,D)) & m1_subset_1(k1_rcomp_1(B,C),k1_zfmisc_1(u1_struct_0(k4_topmetr(A,D)))) ) ) ) ) ) ) ), file(topreala,t44_topreala), []). fof(t47_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,k2_rcomp_1(A,B)) <=> ( ~ r1_xreal_0(C,A) & ~ r1_xreal_0(B,C) ) ) ) ) ) ), file(rcomp_1,t47_rcomp_1), []). fof(t48_rcomp_1,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,k1_rcomp_1(A,B)) <=> ( r1_xreal_0(A,C) & r1_xreal_0(C,B) ) ) ) ) ) ), file(rcomp_1,t48_rcomp_1), []). fof(t4_rcomp_1,axiom,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_numbers)) => ( ( r1_tarski(A,B) & v1_seq_4(B) ) => v1_seq_4(A) ) ) ) ), file(rcomp_1,t4_rcomp_1), []). fof(t4_rcomp_2,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r2_hidden(A,k2_rcomp_2(B,C)) <=> ( ~ r1_xreal_0(A,B) & r1_xreal_0(A,C) ) ) ) ) ) ), file(rcomp_2,t4_rcomp_2), []). fof(t4_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), []). fof(t4_subset,axiom,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), []). fof(t54_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_xreal_0(A,B) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))) => ( ( E = k1_rcomp_2(C,D) & r1_xreal_0(D,B) ) => ( r1_xreal_0(C,A) | k1_tops_1(k4_topmetr(A,B),E) = k2_rcomp_1(C,D) ) ) ) ) ) ) ) ) ), file(rcomp_3,t54_rcomp_3), []). fof(t55_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_xreal_0(A,B) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))) => ( ( E = k2_rcomp_2(C,D) & r1_xreal_0(A,C) ) => ( r1_xreal_0(B,D) | k1_tops_1(k4_topmetr(A,B),E) = k2_rcomp_1(C,D) ) ) ) ) ) ) ) ) ), file(rcomp_3,t55_rcomp_3), []). fof(t55_tops_1,axiom,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( ( v3_pre_topc(D,B) => k1_tops_1(B,D) = D ) & ( k1_tops_1(A,C) = C => v3_pre_topc(C,A) ) ) ) ) ) ) ), file(tops_1,t55_tops_1), []). fof(t56_rcomp_3,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_topmetr(A,B)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_numbers)) => ( C = D => ( v2_connsp_1(C,k4_topmetr(A,B)) <=> v1_jct_misc(D) ) ) ) ) ) ) ), file(rcomp_3,t56_rcomp_3), []). fof(t5_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), []). fof(t5_subset,axiom,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), []). fof(t6_boole,axiom,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), []). fof(t6_real,axiom,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), []). fof(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), []).