fof(t8_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( ( v5_seqm_3(C) & m1_borsuk_2(C,A,B,B) ) => k2_borsuk_2(A,B,B,C) = C ) ) ) ), inference(mizar_proof,[status(thm)],[d1_funct_2,d1_funct_2,t83_borsuk_1,l17_borsuk_2,d6_borsuk_2,d5_seqm_3,t9_funct_1]), [file(borsuk_2,t8_borsuk_2),interesting(1.00)]). fof(t15_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(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) => ( r2_borsuk_2(A,B,C) => r3_borsuk_2(A,B,C,D,D) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t10_mcart_1,t7_funct_2,s1_funct_2,d5_borsuk_1,d7_borsuk_2,d5_borsuk_1,t10_mcart_1,d2_borsuk_2,d2_borsuk_1,d1_connsp_2,t118_zfmisc_1,d5_borsuk_1,t47_borsuk_1,t102_zfmisc_1,t43_tops_1,t10_mcart_1,t11_mcart_1,d1_connsp_2,d3_tarski,d12_funct_1,d1_funct_2,d1_funct_2,t10_mcart_1,d12_funct_1,l1_borsuk_2,t106_zfmisc_1,t7_mcart_1,t7_mcart_1,d2_borsuk_2,l1_borsuk_2,t106_zfmisc_1,t7_mcart_1,d2_borsuk_1]), [file(borsuk_2,t15_borsuk_2),interesting(0.97)]). fof(l17_borsuk_2,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( r2_hidden(A,u1_struct_0(k5_topmetr)) => r2_hidden(k5_real_1(1,A),u1_struct_0(k5_topmetr)) ) ) ), inference(mizar_proof,[status(thm)],[l1_borsuk_2,l16_borsuk_2,l1_borsuk_2]), [file(borsuk_2,l17_borsuk_2),interesting(0.96)]). fof(t5_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_borsuk_2(A) => v1_connsp_1(A) ) ) ), inference(mizar_proof,[status(thm)],[d3_borsuk_2,d1_borsuk_2,l10_borsuk_2,t83_borsuk_1,t51_tops_2,d5_funct_1,t22_treal_1,t2_jordan1]), [file(borsuk_2,t5_borsuk_2),interesting(0.95)]). fof(s1_borsuk_2,theorem,( r1_tarski(k1_card_1(a_0_0_borsuk_2),k1_card_1(f2_s1_borsuk_2)) ), inference(mizar_proof,[status(thm)],[s3_funct_1,d3_tarski,d5_funct_1,t28_card_1]), [file(borsuk_2,s1_borsuk_2),interesting(0.84)]). fof(t7_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( ( v5_seqm_3(C) & m1_borsuk_2(C,A,B,B) ) => k1_borsuk_2(A,B,B,B,C,C) = C ) ) ) ), inference(mizar_proof,[status(thm)],[d1_funct_2,d1_funct_2,t83_borsuk_1,d1_rcomp_1,t121_real_2,t70_xreal_1,t83_borsuk_1,d1_rcomp_1,d5_seqm_3,d5_borsuk_2,t66_xreal_1,t21_xreal_1,t66_xreal_1,t92_real_1,t83_borsuk_1,d1_rcomp_1,d5_seqm_3,d5_borsuk_2,t9_funct_1]), [file(borsuk_2,t7_borsuk_2),interesting(0.82)]). fof(l16_borsuk_2,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( ( r1_xreal_0(0,A) & r1_xreal_0(A,1) ) => ( r1_xreal_0(0,k5_real_1(1,A)) & r1_xreal_0(k5_real_1(1,A),1) ) ) ) ), inference(mizar_proof,[status(thm)],[t92_real_1]), [file(borsuk_2,l16_borsuk_2),interesting(0.82)]). fof(l10_borsuk_2,theorem, ( r2_hidden(0,k1_rcomp_1(0,1)) & r2_hidden(1,k1_rcomp_1(0,1)) ), inference(mizar_proof,[status(thm)],[d1_rcomp_1]), [file(borsuk_2,l10_borsuk_2),interesting(0.79)]). fof(l25_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ( ( v3_compts_1(A) & v3_compts_1(B) ) => v3_compts_1(k6_borsuk_1(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_compts_1,d5_borsuk_1,d2_zfmisc_1,d2_zfmisc_1,d4_compts_1,t53_tops_1,t46_borsuk_1,t106_zfmisc_1,t127_zfmisc_1,d4_compts_1,t53_tops_1,t46_borsuk_1,t106_zfmisc_1,t127_zfmisc_1]), [file(borsuk_2,l25_borsuk_2),interesting(0.77)]). fof(l1_borsuk_2,theorem,( ! [A] : ( v1_xreal_0(A) => ( ( r1_xreal_0(0,A) & r1_xreal_0(A,1) ) <=> r2_hidden(A,u1_struct_0(k5_topmetr)) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d1_rcomp_1,t83_borsuk_1,t83_borsuk_1]), [file(borsuk_2,l1_borsuk_2),interesting(0.77)]). fof(t14_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ( ( v1_urysohn1(A) & v1_urysohn1(B) ) => v1_urysohn1(k6_borsuk_1(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_urysohn1,d5_borsuk_1,d2_zfmisc_1,d2_zfmisc_1,d9_urysohn1,t53_tops_1,t46_borsuk_1,t106_zfmisc_1,t106_zfmisc_1,d9_urysohn1,t53_tops_1,t46_borsuk_1,t106_zfmisc_1,t106_zfmisc_1]), [file(borsuk_2,t14_borsuk_2),interesting(0.76)]). fof(t12_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( ~ v3_struct_0(D) & v2_pre_topc(D) & l1_pre_topc(D) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(C)) & v5_pre_topc(E,A,C) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),u1_struct_0(D)) & v5_pre_topc(F,B,D) & m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) ) => v5_pre_topc(k3_borsuk_2(A,C,B,D,E,F),k6_borsuk_1(A,B),k6_borsuk_1(C,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t11_borsuk_2,t55_tops_2]), [file(borsuk_2,t12_borsuk_2),interesting(0.68)]). fof(t6_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( ( v5_seqm_3(C) & m1_borsuk_2(C,A,B,B) ) => C = k3_borsuk_1(k5_topmetr,A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_borsuk_1,d1_funct_2,d1_funct_2,d2_borsuk_2,t83_borsuk_1,d1_rcomp_1,d5_seqm_3,t13_funcop_1,t9_funct_1]), [file(borsuk_2,t6_borsuk_2),interesting(0.68)]). fof(t3_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v3_tops_2(C,A,B) => v1_t_0topsp(k2_tops_2(A,B,C),B,A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d5_tops_2,d2_t_0topsp,t68_tops_2,t55_tops_2]), [file(borsuk_2,t3_borsuk_2),interesting(0.59)]). fof(t9_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) ) => k5_pre_topc(A,B,D,k5_setfam_1(u1_struct_0(B),C)) = k5_setfam_1(u1_struct_0(A),k1_weierstr(A,B,D,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t46_funct_2,d4_tarski,d1_weierstr,t46_funct_2,d4_tarski,d10_xboole_0,d3_tarski,d4_tarski,d1_weierstr,d13_funct_1,d4_tarski,d1_funct_2,d13_funct_1]), [file(borsuk_2,t9_borsuk_2),interesting(0.57)]). fof(t11_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( ~ v3_struct_0(D) & v2_pre_topc(D) & l1_pre_topc(D) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(C)) & v5_pre_topc(E,A,C) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),u1_struct_0(D)) & v5_pre_topc(F,B,D) & m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) ) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,D)))) => ( v3_pre_topc(G,k6_borsuk_1(C,D)) => v3_pre_topc(k5_pre_topc(k6_borsuk_1(A,B),k6_borsuk_1(C,D),k3_borsuk_2(A,C,B,D,E,F),G),k6_borsuk_1(A,B)) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t54_borsuk_1,t9_borsuk_2,d1_weierstr,t10_borsuk_2,d1_tops_2,t26_tops_2]), [file(borsuk_2,t11_borsuk_2),interesting(0.57)]). fof(t4_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) & v5_pre_topc(C,k5_topmetr,A) & k1_funct_1(C,0) = B & k1_funct_1(C,1) = B ) ) ) ), inference(mizar_proof,[status(thm)],[d3_borsuk_1,d17_borsuk_1,d18_borsuk_1,t13_funcop_1]), [file(borsuk_2,t4_borsuk_2),interesting(0.32)]). fof(t10_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( ~ v3_struct_0(D) & v2_pre_topc(D) & l1_pre_topc(D) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(C)) & v5_pre_topc(E,A,C) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(C)) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(B),u1_struct_0(D)) & v5_pre_topc(F,B,D) & m2_relset_1(F,u1_struct_0(B),u1_struct_0(D)) ) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,D)))) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(C,D)))) => ( r2_hidden(H,k11_borsuk_1(C,D,G)) => v3_pre_topc(k5_pre_topc(k6_borsuk_1(A,B),k6_borsuk_1(C,D),k3_borsuk_2(A,C,B,D,E,F),H),k6_borsuk_1(A,B)) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d6_borsuk_1,t94_funct_3,t55_tops_2,t46_borsuk_1]), [file(borsuk_2,t10_borsuk_2),interesting(0.30)]). fof(d4_compts_1,definition,( ! [A] : ( l1_pre_topc(A) => ( v3_compts_1(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ~ ( B != C & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v3_pre_topc(D,A) & v3_pre_topc(E,A) & r2_hidden(B,D) & r2_hidden(C,E) & r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ), file(compts_1,d4_compts_1), [interesting(0.00)]). fof(d5_borsuk_1,definition,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_pre_topc(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ( C = k6_borsuk_1(A,B) <=> ( u1_struct_0(C) = k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B)) & u1_pre_topc(C) = a_3_0_borsuk_1(A,B,C) ) ) ) ) ) ), file(borsuk_1,d5_borsuk_1), [interesting(0.00)]). fof(d2_zfmisc_1,definition,( ! [A,B,C] : ( C = k2_zfmisc_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E,F] : ( r2_hidden(E,A) & r2_hidden(F,B) & D = k4_tarski(E,F) ) ) ) ), file(zfmisc_1,d2_zfmisc_1), [interesting(0.00)]). fof(t53_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v3_pre_topc(k2_pre_topc(A),A) ) ), file(tops_1,t53_tops_1), [interesting(0.00)]). fof(t46_borsuk_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( ( v3_pre_topc(C,A) & v3_pre_topc(D,B) ) => v3_pre_topc(k7_borsuk_1(A,B,C,D),k6_borsuk_1(A,B)) ) ) ) ) ) ), file(borsuk_1,t46_borsuk_1), [interesting(0.00)]). fof(t106_zfmisc_1,theorem,( ! [A,B,C,D] : ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D)) <=> ( r2_hidden(A,C) & r2_hidden(B,D) ) ) ), file(zfmisc_1,t106_zfmisc_1), [interesting(0.00)]). fof(t127_zfmisc_1,theorem,( ! [A,B,C,D] : ( ( r1_xboole_0(A,B) | r1_xboole_0(C,D) ) => r1_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ), file(zfmisc_1,t127_zfmisc_1), [interesting(0.00)]). fof(s3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s3_funct_1 & ! [B] : ( r2_hidden(B,f1_s3_funct_1) => k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) ), file(funct_1,s3_funct_1), [interesting(0.00)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.00)]). fof(d5_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), [interesting(0.00)]). fof(t28_card_1,theorem,( ! [A,B] : ( r1_tarski(k1_card_1(A),k1_card_1(B)) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & k1_relat_1(C) = B & r1_tarski(A,k2_relat_1(C)) ) ) ), file(card_1,t28_card_1), [interesting(0.00)]). fof(t54_borsuk_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) => ( v3_pre_topc(C,k6_borsuk_1(A,B)) => C = k5_setfam_1(u1_struct_0(k6_borsuk_1(A,B)),k11_borsuk_1(A,B,C)) ) ) ) ) ), file(borsuk_1,t54_borsuk_1), [interesting(0.00)]). fof(t46_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( B != k1_xboole_0 => ! [E] : ( r2_hidden(E,k10_relat_1(D,C)) <=> ( r2_hidden(E,A) & r2_hidden(k1_funct_1(D,E),C) ) ) ) ) ), file(funct_2,t46_funct_2), [interesting(0.00)]). fof(d4_tarski,definition,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), [interesting(0.00)]). fof(d1_weierstr,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( E = k1_weierstr(A,B,C,D) <=> ! [F] : ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(F,E) <=> ? [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(B))) & r2_hidden(G,D) & F = k5_pre_topc(A,B,C,G) ) ) ) ) ) ) ) ) ) ), file(weierstr,d1_weierstr), [interesting(0.00)]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.00)]). fof(d13_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k10_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,k1_relat_1(A)) & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ), file(funct_1,d13_funct_1), [interesting(0.00)]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [interesting(0.00)]). fof(d6_borsuk_1,definition,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B)))) => k11_borsuk_1(A,B,C) = a_3_1_borsuk_1(A,B,C) ) ) ) ), file(borsuk_1,d6_borsuk_1), [interesting(0.00)]). fof(t94_funct_3,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) ) => k10_relat_1(k15_funct_3(C,D),k2_zfmisc_1(A,B)) = k2_zfmisc_1(k10_relat_1(C,A),k10_relat_1(D,B)) ) ) ), file(funct_3,t94_funct_3), [interesting(0.00)]). fof(t55_tops_2,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v5_pre_topc(C,A,B) <=> ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( v3_pre_topc(D,B) => v3_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ), file(tops_2,t55_tops_2), [interesting(0.00)]). fof(d1_tops_2,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_tops_2(B,A) <=> ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(C,B) => v3_pre_topc(C,A) ) ) ) ) ) ), file(tops_2,d1_tops_2), [interesting(0.00)]). fof(t26_tops_2,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_tops_2(B,A) => v3_pre_topc(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_2,t26_tops_2), [interesting(0.00)]). fof(t13_borsuk_2,theorem,( $true ), file(borsuk_2,t13_borsuk_2), [interesting(0.00)]). fof(d9_urysohn1,definition,( ! [A] : ( l1_pre_topc(A) => ( v1_urysohn1(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ~ ( B != C & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v3_pre_topc(D,A) & v3_pre_topc(E,A) & r2_hidden(B,D) & ~ r2_hidden(C,D) & r2_hidden(C,E) & ~ r2_hidden(B,E) ) ) ) ) ) ) ) ) ), file(urysohn1,d9_urysohn1), [interesting(0.00)]). fof(t10_mcart_1,theorem,( ! [A,B,C] : ( r2_hidden(A,k2_zfmisc_1(B,C)) => ( r2_hidden(k1_mcart_1(A),B) & r2_hidden(k2_mcart_1(A),C) ) ) ), file(mcart_1,t10_mcart_1), [interesting(0.00)]). fof(t7_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | r2_hidden(k1_funct_1(D,C),B) ) ) ) ), file(funct_2,t7_funct_2), [interesting(0.00)]). fof(s1_funct_2,theorem, ( ! [A] : ~ ( r2_hidden(A,f1_s1_funct_2) & ! [B] : ~ ( r2_hidden(B,f2_s1_funct_2) & p1_s1_funct_2(A,B) ) ) => ? [A] : ( v1_funct_1(A) & v1_funct_2(A,f1_s1_funct_2,f2_s1_funct_2) & m2_relset_1(A,f1_s1_funct_2,f2_s1_funct_2) & ! [B] : ( r2_hidden(B,f1_s1_funct_2) => p1_s1_funct_2(B,k1_funct_1(A,B)) ) ) ), file(funct_2,s1_funct_2), [interesting(0.00)]). fof(d7_borsuk_2,definition,( ! [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), [interesting(0.00)]). fof(d2_borsuk_2,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r1_borsuk_2(A,B,C) => ! [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,d2_borsuk_2), [interesting(0.00)]). fof(d2_borsuk_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v5_pre_topc(C,A,B) <=> ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_connsp_2(E,B,k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,D)) => ? [F] : ( m1_connsp_2(F,A,D) & r1_tarski(k4_pre_topc(A,B,C,F),E) ) ) ) ) ) ) ) ), file(borsuk_1,d2_borsuk_1), [interesting(0.00)]). fof(d1_connsp_2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( m1_connsp_2(C,A,B) <=> r2_hidden(B,k1_tops_1(A,C)) ) ) ) ) ), file(connsp_2,d1_connsp_2), [interesting(0.00)]). fof(t118_zfmisc_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => ( r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C)) & r1_tarski(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ) ), file(zfmisc_1,t118_zfmisc_1), [interesting(0.00)]). fof(t47_borsuk_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => k1_tops_1(k6_borsuk_1(A,B),k7_borsuk_1(A,B,C,D)) = k7_borsuk_1(A,B,k1_tops_1(A,C),k1_tops_1(B,D)) ) ) ) ) ), file(borsuk_1,t47_borsuk_1), [interesting(0.00)]). fof(t102_zfmisc_1,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,k2_zfmisc_1(B,C)) & ! [D,E] : k4_tarski(D,E) != A ) ), file(zfmisc_1,t102_zfmisc_1), [interesting(0.00)]). fof(t43_tops_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => k1_tops_1(A,k2_pre_topc(A)) = k2_pre_topc(A) ) ), file(tops_1,t43_tops_1), [interesting(0.00)]). fof(t11_mcart_1,theorem,( ! [A,B,C] : ( ( r2_hidden(k1_mcart_1(A),B) & r2_hidden(k2_mcart_1(A),C) ) => ( ! [D,E] : A != k4_tarski(D,E) | r2_hidden(A,k2_zfmisc_1(B,C)) ) ) ), file(mcart_1,t11_mcart_1), [interesting(0.00)]). fof(d12_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k9_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( r2_hidden(E,k1_relat_1(A)) & r2_hidden(E,B) & D = k1_funct_1(A,E) ) ) ) ) ), file(funct_1,d12_funct_1), [interesting(0.00)]). fof(d1_xreal_0,definition,( ! [A] : ( v1_xreal_0(A) <=> r2_hidden(A,k1_numbers) ) ), file(xreal_0,d1_xreal_0), [interesting(0.00)]). fof(d1_rcomp_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k1_rcomp_1(A,B) = a_2_0_rcomp_1(A,B) ) ) ), file(rcomp_1,d1_rcomp_1), [interesting(0.00)]). fof(t83_borsuk_1,theorem,( u1_struct_0(k22_borsuk_1) = k1_rcomp_1(0,1) ), file(borsuk_1,t83_borsuk_1), [interesting(0.00)]). fof(t7_mcart_1,theorem,( ! [A,B] : ( k1_mcart_1(k4_tarski(A,B)) = A & k2_mcart_1(k4_tarski(A,B)) = B ) ), file(mcart_1,t7_mcart_1), [interesting(0.00)]). fof(d1_funct_4,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( C = k1_funct_4(A,B) <=> ( k1_relat_1(C) = k2_xboole_0(k1_relat_1(A),k1_relat_1(B)) & ! [D] : ( r2_hidden(D,k2_xboole_0(k1_relat_1(A),k1_relat_1(B))) => ( ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(C,D) = k1_funct_1(B,D) ) & ( ~ r2_hidden(D,k1_relat_1(B)) => k1_funct_1(C,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) ) ), file(funct_4,d1_funct_4), [interesting(0.00)]). fof(t12_relset_1,theorem,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) ) ), file(relset_1,t12_relset_1), [interesting(0.00)]). fof(t18_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(k2_relat_1(k1_funct_4(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ), file(funct_4,t18_funct_4), [interesting(0.00)]). fof(t8_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), [interesting(0.00)]). fof(t1_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(B,C) ) => r1_tarski(A,C) ) ), file(xboole_1,t1_xboole_1), [interesting(0.00)]). fof(t4_funct_2,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(k2_relat_1(B),A) => ( v1_funct_1(B) & v1_funct_2(B,k1_relat_1(B),A) & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ), file(funct_2,t4_funct_2), [interesting(0.00)]). fof(t167_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => r1_tarski(k10_relat_1(B,A),k1_relat_1(B)) ) ), file(relat_1,t167_relat_1), [interesting(0.00)]). fof(t28_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), [interesting(0.00)]). fof(t23_xboole_1,theorem,( ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ), file(xboole_1,t23_xboole_1), [interesting(0.00)]). fof(d3_xboole_0,definition,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), [interesting(0.00)]). fof(t14_funct_4,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(B,A) ) ) ) ), file(funct_4,t14_funct_4), [interesting(0.00)]). fof(t12_funct_4,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ~ r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(C,A) ) ) ) ), file(funct_4,t12_funct_4), [interesting(0.00)]). fof(d2_xboole_0,definition,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), [interesting(0.00)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.00)]). fof(t7_xboole_1,theorem,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), [interesting(0.00)]). fof(d12_pre_topc,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v5_pre_topc(C,A,B) <=> ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( v4_pre_topc(D,B) => v4_pre_topc(k5_pre_topc(A,B,C,D),A) ) ) ) ) ) ) ), file(pre_topc,d12_pre_topc), [interesting(0.00)]). fof(t17_compts_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v2_compts_1(A) & v4_pre_topc(B,A) ) => v6_compts_1(B,A) ) ) ) ), file(compts_1,t17_compts_1), [interesting(0.00)]). fof(t42_borsuk_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( C = D => ( v6_compts_1(C,A) <=> v6_compts_1(D,B) ) ) ) ) ) ) ), file(borsuk_1,t42_borsuk_1), [interesting(0.00)]). fof(t19_compts_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v6_compts_1(B,A) & v6_compts_1(C,A) ) => v6_compts_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ), file(compts_1,t19_compts_1), [interesting(0.00)]). fof(t16_compts_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v3_compts_1(A) & v6_compts_1(B,A) ) => v4_pre_topc(B,A) ) ) ) ), file(compts_1,t16_compts_1), [interesting(0.00)]). fof(t1_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( ~ v3_struct_0(C) & v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( ~ v3_struct_0(D) & v2_pre_topc(D) & l1_pre_topc(D) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(C),u1_struct_0(B)) & m2_relset_1(F,u1_struct_0(C),u1_struct_0(B)) ) => ~ ( m1_pre_topc(A,D) & m1_pre_topc(C,D) & k2_xboole_0(k2_pre_topc(A),k2_pre_topc(C)) = k2_pre_topc(D) & v2_compts_1(A) & v2_compts_1(C) & v3_compts_1(D) & v5_pre_topc(E,A,B) & v5_pre_topc(F,C,B) & ! [G] : ( r2_hidden(G,k3_xboole_0(k2_pre_topc(A),k2_pre_topc(C))) => k1_funct_1(E,G) = k1_funct_1(F,G) ) & ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(D),u1_struct_0(B)) & m2_relset_1(G,u1_struct_0(D),u1_struct_0(B)) ) => ~ ( G = k1_funct_4(E,F) & v5_pre_topc(G,D,B) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_funct_2,d1_funct_2,d1_funct_4,t12_relset_1,t18_funct_4,t8_xboole_1,t1_xboole_1,t4_funct_2,d1_funct_4,t167_relat_1,t28_xboole_1,t23_xboole_1,d3_xboole_0,t14_funct_4,t12_funct_4,d3_xboole_0,d13_funct_1,d13_funct_1,d13_funct_1,d2_xboole_0,d13_funct_1,d3_xboole_0,t2_tarski,d3_xboole_0,d13_funct_1,t14_funct_4,d13_funct_1,d13_funct_1,t14_funct_4,d2_xboole_0,d13_funct_1,d3_xboole_0,t2_tarski,t7_xboole_1,t1_xboole_1,t7_xboole_1,t1_xboole_1,d12_pre_topc,t17_compts_1,t42_borsuk_1,t19_compts_1,t16_compts_1,d12_pre_topc]), [file(borsuk_2,t1_borsuk_2),interesting(0.00)]). fof(t2_borsuk_2,theorem,( $true ), file(borsuk_2,t2_borsuk_2), [interesting(0.00)]). fof(d5_tops_2,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v3_tops_2(C,A,B) <=> ( k1_relat_1(C) = k2_pre_topc(A) & k2_relat_1(C) = k2_pre_topc(B) & v2_funct_1(C) & v5_pre_topc(C,A,B) & v5_pre_topc(k2_tops_2(A,B,C),B,A) ) ) ) ) ) ), file(tops_2,d5_tops_2), [interesting(0.00)]). fof(d2_t_0topsp,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( v1_t_0topsp(C,A,B) <=> ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( v3_pre_topc(D,A) => v3_pre_topc(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ) ), file(t_0topsp,d2_t_0topsp), [interesting(0.00)]). fof(t68_tops_2,theorem,( ! [A] : ( l1_struct_0(A) => ! [B] : ( l1_struct_0(B) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( ( k2_relat_1(C) = k2_pre_topc(B) & v2_funct_1(C) ) => k5_pre_topc(A,B,C,D) = k4_pre_topc(B,A,k2_tops_2(A,B,C),D) ) ) ) ) ) ), file(tops_2,t68_tops_2), [interesting(0.00)]). fof(d3_borsuk_1,definition,( ! [A] : ( l1_struct_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_struct_0(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => k3_borsuk_1(A,B,C) = k1_borsuk_1(u1_struct_0(B),u1_struct_0(A),C) ) ) ) ), file(borsuk_1,d3_borsuk_1), [interesting(0.00)]). fof(d17_borsuk_1,definition,( k23_borsuk_1 = 0 ), file(borsuk_1,d17_borsuk_1), [interesting(0.00)]). fof(d18_borsuk_1,definition,( k24_borsuk_1 = 1 ), file(borsuk_1,d18_borsuk_1), [interesting(0.00)]). fof(t13_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(B,A) => k1_funct_1(k2_funcop_1(A,C),B) = C ) ), file(funcop_1,t13_funcop_1), [interesting(0.00)]). fof(d3_borsuk_2,definition,( ! [A] : ( l1_pre_topc(A) => ( v1_borsuk_2(A) <=> ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => r1_borsuk_2(A,B,C) ) ) ) ) ), file(borsuk_2,d3_borsuk_2), [interesting(0.00)]). fof(d1_borsuk_2,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r1_borsuk_2(A,B,C) <=> ? [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)) & v5_pre_topc(D,k5_topmetr,A) & k1_funct_1(D,0) = B & k1_funct_1(D,1) = C ) ) ) ) ) ), file(borsuk_2,d1_borsuk_2), [interesting(0.00)]). fof(t51_tops_2,theorem,( ! [A] : ( l1_struct_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_struct_0(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ( k1_relat_1(C) = k2_pre_topc(A) & r1_tarski(k2_relat_1(C),k2_pre_topc(B)) ) ) ) ) ), file(tops_2,t51_tops_2), [interesting(0.00)]). fof(t22_treal_1,theorem,( v1_connsp_1(k22_borsuk_1) ), file(treal_1,t22_treal_1), [interesting(0.00)]). fof(t2_jordan1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ? [D] : ( ~ v3_struct_0(D) & v2_pre_topc(D) & l1_pre_topc(D) & v1_connsp_1(D) & ? [E] : ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(D),u1_struct_0(A)) & m2_relset_1(E,u1_struct_0(D),u1_struct_0(A)) & v5_pre_topc(E,D,A) & r2_hidden(B,k2_relat_1(E)) & r2_hidden(C,k2_relat_1(E)) ) ) ) ) => v1_connsp_1(A) ) ) ), file(jordan1,t2_jordan1), [interesting(0.00)]). fof(d5_seqm_3,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v5_seqm_3(A) <=> ! [B,C] : ( ( r2_hidden(B,k1_relat_1(A)) & r2_hidden(C,k1_relat_1(A)) ) => k1_funct_1(A,B) = k1_funct_1(A,C) ) ) ) ), file(seqm_3,d5_seqm_3), [interesting(0.00)]). fof(t9_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( k1_relat_1(A) = k1_relat_1(B) & ! [C] : ( r2_hidden(C,k1_relat_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) => A = B ) ) ) ), file(funct_1,t9_funct_1), [interesting(0.00)]). fof(t121_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ( r1_xreal_0(0,A) & r1_xreal_0(0,B) ) | ( r1_xreal_0(A,0) & r1_xreal_0(B,0) ) ) => r1_xreal_0(0,k3_xcmplx_0(A,B)) ) ) ) ), file(real_2,t121_real_2), [interesting(0.00)]). fof(t70_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(C,B) & r1_xreal_0(k3_xcmplx_0(C,A),k3_xcmplx_0(B,A)) ) ) ) ) ), file(xreal_1,t70_xreal_1), [interesting(0.00)]). fof(d5_borsuk_2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(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_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_borsuk_2(E,A,B,C) => ! [F] : ( m1_borsuk_2(F,A,C,D) => ( ( r2_borsuk_2(A,B,C) & r2_borsuk_2(A,C,D) ) => ! [G] : ( m1_borsuk_2(G,A,B,D) => ( G = k1_borsuk_2(A,B,C,D,E,F) <=> ! [H] : ( m1_subset_1(H,u1_struct_0(k5_topmetr)) => ( ( r1_xreal_0(H,k6_real_1(1,2)) => k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),G,H) = k1_funct_1(E,k3_xcmplx_0(2,H)) ) & ( r1_xreal_0(k6_real_1(1,2),H) => k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),G,H) = k1_funct_1(F,k6_xcmplx_0(k3_xcmplx_0(2,H),1)) ) ) ) ) ) ) ) ) ) ) ) ) ), file(borsuk_2,d5_borsuk_2), [interesting(0.00)]). fof(t66_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(0,C) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t66_xreal_1), [interesting(0.00)]). fof(t21_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(k2_xcmplx_0(A,B),C) <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ), file(xreal_1,t21_xreal_1), [interesting(0.00)]). fof(t92_real_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k6_xcmplx_0(A,D),k6_xcmplx_0(B,C)) ) & ~ ( ( ( ~ r1_xreal_0(B,A) & r1_xreal_0(C,D) ) | ( r1_xreal_0(A,B) & ~ r1_xreal_0(D,C) ) ) & r1_xreal_0(k6_xcmplx_0(B,C),k6_xcmplx_0(A,D)) ) ) ) ) ) ) ), file(real_1,t92_real_1), [interesting(0.00)]). fof(d6_borsuk_2,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(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) => ( r2_borsuk_2(A,B,C) => ! [E] : ( m1_borsuk_2(E,A,C,B) => ( E = k2_borsuk_2(A,B,C,D) <=> ! [F] : ( m1_subset_1(F,u1_struct_0(k5_topmetr)) => k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(A),E,F) = k1_funct_1(D,k6_xcmplx_0(1,F)) ) ) ) ) ) ) ) ) ), file(borsuk_2,d6_borsuk_2), [interesting(0.00)]).