fof(l14_borsuk_5,theorem,( v1_borsuk_2(k3_topmetr) ), inference(mizar_proof,[status(thm)],[d3_borsuk_2,t24_topmetr,t48_rcomp_1,t26_topmetr,t27_topmetr,t11_treal_1,t35_borsuk_1,t9_funct_2,t27_topmetr,d1_borsuk_2,t7_topmetr,d1_treal_1,t12_treal_1,d1_treal_1,d2_treal_1,t12_treal_1,d2_treal_1,t24_topmetr,t48_rcomp_1,t26_topmetr,t27_topmetr,t11_treal_1,t35_borsuk_1,t9_funct_2,t27_topmetr,t7_topmetr,d1_treal_1,t12_treal_1,d1_treal_1,d2_treal_1,t12_treal_1,d2_treal_1,d1_borsuk_2,d2_borsuk_2,d1_borsuk_2,t11_borsuk_5,d1_borsuk_2,d2_borsuk_2]), [file(borsuk_5,l14_borsuk_5),interesting(1.00)]). fof(t10_borsuk_5,theorem,( ! [A,B,C,D,E,F,G] : ( r2_borsuk_5(A,B,C,D,E,F,G) => r2_borsuk_5(A,B,E,C,F,G,D) ) ), inference(mizar_proof,[status(thm)],[d2_borsuk_5,d2_borsuk_5]), [file(borsuk_5,t10_borsuk_5),interesting(0.92)]). fof(t111_borsuk_5,theorem,( ! [A] : ( ( v2_connsp_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r2_hidden(B,A) & r2_hidden(C,A) ) => r1_tarski(k1_rcomp_1(B,C),A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_rcomp_1,t110_borsuk_5]), [file(borsuk_5,t111_borsuk_5),interesting(0.92)]). fof(t9_borsuk_5,theorem,( ! [A,B,C,D,E,F,G] : ( r2_borsuk_5(A,B,C,D,E,F,G) => r2_borsuk_5(G,A,B,C,D,E,F) ) ), inference(mizar_proof,[status(thm)],[d2_borsuk_5,d2_borsuk_5]), [file(borsuk_5,t9_borsuk_5),interesting(0.92)]). fof(t112_borsuk_5,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k3_topmetr) & v6_compts_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ( ~ v1_xboole_0(A) & v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t16_compts_1,t79_topreal6,t107_borsuk_5,t31_rcomp_1,t30_rcomp_1,t111_borsuk_5,t109_borsuk_5,d3_seq_4,t24_seq_4,d1_integra1]), [file(borsuk_5,t112_borsuk_5),interesting(0.90)]). fof(t8_borsuk_5,theorem,( ! [A,B,C,D,E,F,G] : ( ( r1_borsuk_5(A,B,C,D,E,F) & r1_xboole_0(k4_enumset1(A,B,C,D,E,F),k1_tarski(G)) ) => r2_borsuk_5(A,B,C,D,E,F,G) ) ), inference(mizar_proof,[status(thm)],[d1_borsuk_5,t54_zfmisc_1,d4_enumset1,d2_borsuk_5]), [file(borsuk_5,t8_borsuk_5),interesting(0.89)]). fof(t76_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k7_pscomp_1(k4_limfunc1(A)) = k3_limfunc1(A) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t75_borsuk_5,t80_topreal6]), [file(borsuk_5,t76_borsuk_5),interesting(0.88)]). fof(t78_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k7_pscomp_1(k12_prob_1(A)) = k2_limfunc1(A) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t77_borsuk_5,t80_topreal6]), [file(borsuk_5,t78_borsuk_5),interesting(0.87)]). fof(t110_borsuk_5,theorem,( ! [A] : ( ( v2_connsp_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(B,C) & r1_xreal_0(C,D) & r2_hidden(B,A) & r2_hidden(D,A) ) => r2_hidden(C,A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t40_zfmisc_1,t18_borsuk_5,t24_topmetr,t79_borsuk_5,t17_borsuk_5,t14_borsuk_5,t17_connsp_1,d5_real_1]), [file(borsuk_5,t110_borsuk_5),interesting(0.86)]). fof(t4_borsuk_5,theorem,( ! [A,B,C,D,E,F] : ( r1_borsuk_5(A,B,C,D,E,F) => k4_card_1(k4_enumset1(A,B,C,D,E,F)) = 6 ) ), inference(mizar_proof,[status(thm)],[d1_borsuk_5,d1_necklace,t1_necklace,d3_enumset1,t55_enumset1,t54_card_2]), [file(borsuk_5,t4_borsuk_5),interesting(0.82)]). fof(t5_borsuk_5,theorem,( ! [A,B,C,D,E,F,G] : ( r2_borsuk_5(A,B,C,D,E,F,G) => k4_card_1(k5_enumset1(A,B,C,D,E,F,G)) = 7 ) ), inference(mizar_proof,[status(thm)],[d2_borsuk_5,d1_borsuk_5,t4_borsuk_5,d4_enumset1,t61_enumset1,t54_card_2]), [file(borsuk_5,t5_borsuk_5),interesting(0.82)]). fof(t50_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & ! [C] : ( ( v1_xreal_0(C) & ~ v1_rat_1(C) ) => ~ ( ~ r1_xreal_0(C,A) & ~ r1_xreal_0(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t47_borsuk_5,t42_borsuk_5,t38_borsuk_5,t69_int_1,l58_borsuk_5,t2_xreal_1,l57_borsuk_5,t2_xreal_1]), [file(borsuk_5,t50_borsuk_5),interesting(0.81)]). fof(t113_borsuk_5,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k3_topmetr) & v6_compts_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ? [B] : ( v1_xreal_0(B) & ? [C] : ( v1_xreal_0(C) & r1_xreal_0(B,C) & A = k1_rcomp_1(B,C) ) ) ) ), inference(mizar_proof,[status(thm)],[t112_borsuk_5,t5_integra1,t53_borsuk_4,t15_rcomp_1,t53_borsuk_4,t5_integra1]), [file(borsuk_5,t113_borsuk_5),interesting(0.81)]). fof(t41_borsuk_5,theorem,( ! [A] : ( v1_rat_1(A) => ! [B] : ( ( v1_xreal_0(B) & ~ v1_rat_1(B) ) => ~ v1_rat_1(k6_xcmplx_0(B,A)) ) ) ), inference(mizar_proof,[status(thm)],[t38_borsuk_5]), [file(borsuk_5,t41_borsuk_5),interesting(0.73)]). fof(t38_borsuk_5,theorem,( ! [A] : ( v1_rat_1(A) => ! [B] : ( ( v1_xreal_0(B) & ~ v1_rat_1(B) ) => ~ v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[t3_rat_1,t3_rat_1,t131_xcmplx_1]), [file(borsuk_5,t38_borsuk_5),interesting(0.72)]). fof(t77_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k12_prob_1(B) => k6_pre_topc(k3_topmetr,A) = k2_limfunc1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[t64_borsuk_5,t24_topmetr,t65_borsuk_5,t68_borsuk_5,t48_pre_topc,l87_borsuk_5,t31_tops_1,t57_borsuk_5,t74_borsuk_5,t2_borsuk_5]), [file(borsuk_5,t77_borsuk_5),interesting(0.71)]). fof(t75_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k4_limfunc1(B) => k6_pre_topc(k3_topmetr,A) = k3_limfunc1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[t63_borsuk_5,t24_topmetr,t66_borsuk_5,t67_borsuk_5,t48_pre_topc,l86_borsuk_5,t31_tops_1,t57_borsuk_5,t71_borsuk_5,t2_borsuk_5]), [file(borsuk_5,t75_borsuk_5),interesting(0.71)]). fof(t84_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => r1_xboole_0(k3_borsuk_5(A,B),k2_borsuk_5(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,d1_xreal_0,t52_borsuk_5,t53_borsuk_5]), [file(borsuk_5,t84_borsuk_5),interesting(0.71)]). fof(t42_borsuk_5,theorem,( ! [A] : ( v1_rat_1(A) => ! [B] : ( ( v1_xreal_0(B) & ~ v1_rat_1(B) ) => ~ ( A != 0 & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_rat_1,t3_rat_1,t85_xcmplx_1,t90_xcmplx_1]), [file(borsuk_5,t42_borsuk_5),interesting(0.70)]). fof(t37_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ( ~ v1_rat_1(A) <=> r2_hidden(A,k1_borsuk_5) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d2_rat_1,d4_xboole_0,d4_xboole_0,d2_rat_1]), [file(borsuk_5,t37_borsuk_5),interesting(0.70)]). fof(t51_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = k1_borsuk_5 => k6_pre_topc(k3_topmetr,A) = u1_struct_0(k3_topmetr) ) ) ), inference(mizar_proof,[status(thm)],[d10_xboole_0,d3_tarski,d14_metric_1,t24_topmetr,d14_metric_1,d1_xreal_0,t31_xreal_1,t50_borsuk_5,t37_borsuk_5,d14_metric_1,d1_xreal_0,t11_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr]), [file(borsuk_5,t51_borsuk_5),interesting(0.70)]). fof(t54_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_borsuk_5(B,C) => ( r1_xreal_0(C,B) | k6_pre_topc(k3_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_xboole_1,t24_topmetr,t1_xboole_1,t12_numbers,t24_topmetr,t51_pre_topc,t28_xboole_1,d3_tarski,t35_borsuk_5,t36_borsuk_5,d10_xboole_0,d3_tarski,d14_metric_1,t48_rcomp_1,t48_rcomp_1,d14_metric_1,d1_xreal_0,t35_square_1,t3_topreal3,t31_xreal_1,t3_topreal3,t38_square_1,t22_rat_1,d2_rat_1,t2_xreal_1,t2_xreal_1,t47_rcomp_1,d3_xboole_0,d14_metric_1,d1_xreal_0,t11_xreal_1,t35_square_1,t11_xreal_1,t2_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr,l63_borsuk_5,d5_real_1]), [file(borsuk_5,t54_borsuk_5),interesting(0.70)]). fof(l78_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => v3_rcomp_1(k12_prob_1(A)) ) ), inference(mizar_proof,[status(thm)],[t8_fcont_3]), [file(borsuk_5,l78_borsuk_5),interesting(0.69)]). fof(l77_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => v2_rcomp_1(k2_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[t6_fcont_3]), [file(borsuk_5,l77_borsuk_5),interesting(0.69)]). fof(t74_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k12_prob_1(A) != k1_numbers ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t31_xreal_1,t17_borsuk_5]), [file(borsuk_5,t74_borsuk_5),interesting(0.69)]). fof(t97_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k4_subset_1(k1_numbers,k12_prob_1(B),k2_rcomp_2(B,C)) => ( r1_xreal_0(C,B) | k6_pre_topc(k3_topmetr,A) = k2_limfunc1(C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t59_borsuk_5,t50_pre_topc,t77_borsuk_5,t29_borsuk_5]), [file(borsuk_5,t97_borsuk_5),interesting(0.69)]). fof(l26_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k4_xboole_0(k1_numbers,k2_limfunc1(A)) = k4_limfunc1(A) ) ), inference(mizar_proof,[status(thm)],[t24_limfunc1]), [file(borsuk_5,l26_borsuk_5),interesting(0.69)]). fof(l76_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => v3_rcomp_1(k4_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[t7_fcont_3]), [file(borsuk_5,l76_borsuk_5),interesting(0.69)]). fof(l79_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => v2_rcomp_1(k3_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[t5_fcont_3]), [file(borsuk_5,l79_borsuk_5),interesting(0.69)]). fof(l25_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k4_xboole_0(k1_numbers,k12_prob_1(A)) = k3_limfunc1(A) ) ), inference(mizar_proof,[status(thm)],[t24_limfunc1]), [file(borsuk_5,l25_borsuk_5),interesting(0.69)]). fof(t81_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k4_subset_1(k1_numbers,k2_rcomp_1(B,C),k4_limfunc1(C)) => ( r1_xreal_0(C,B) | k6_pre_topc(k3_topmetr,A) = k3_limfunc1(B) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t50_pre_topc,t36_borsuk_5,t75_borsuk_5,t28_borsuk_5]), [file(borsuk_5,t81_borsuk_5),interesting(0.69)]). fof(l27_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k4_xboole_0(k1_numbers,k3_limfunc1(A)) = k12_prob_1(A) ) ), inference(mizar_proof,[status(thm)],[t24_limfunc1]), [file(borsuk_5,l27_borsuk_5),interesting(0.69)]). fof(t68_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k2_limfunc1(A) = k4_subset_1(k1_numbers,k1_seq_4(A),k12_prob_1(A)) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,t16_borsuk_5,d1_tarski,d2_xboole_0,t17_borsuk_5,d2_xboole_0,d5_real_1,d10_xboole_0,d3_tarski,d1_xreal_0,d2_xboole_0,t17_borsuk_5,d1_tarski,t16_borsuk_5]), [file(borsuk_5,t68_borsuk_5),interesting(0.68)]). fof(t71_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k4_limfunc1(A) != k1_numbers ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t14_borsuk_5]), [file(borsuk_5,t71_borsuk_5),interesting(0.68)]). fof(t85_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k4_xboole_0(k1_numbers,k2_borsuk_5(A,B)) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_limfunc1(A),k3_borsuk_5(A,B)),k3_limfunc1(B)) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d4_xboole_0,d1_xreal_0,t16_borsuk_5,d2_xboole_0,d2_xboole_0,t16_borsuk_5,d2_xboole_0,d2_xboole_0,t52_borsuk_5,t53_borsuk_5,d2_xboole_0,d2_xboole_0,t15_borsuk_5,d2_xboole_0,d10_xboole_0,d3_tarski,d1_xreal_0,d2_xboole_0,t16_borsuk_5,t52_borsuk_5,d1_xreal_0,d4_xboole_0,d1_xreal_0,t84_borsuk_5,t3_xboole_0,d4_xboole_0,t15_borsuk_5,t52_borsuk_5,d1_xreal_0,d4_xboole_0,d2_xboole_0,l105_borsuk_5,l104_borsuk_5,l105_borsuk_5,t12_xboole_1,t4_xboole_1]), [file(borsuk_5,t85_borsuk_5),interesting(0.68)]). fof(t44_borsuk_5,theorem,( ! [A] : ( v1_rat_1(A) => ! [B] : ( ( v1_xreal_0(B) & ~ v1_rat_1(B) ) => ~ ( A != 0 & v1_rat_1(k7_xcmplx_0(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t50_xcmplx_1,t42_borsuk_5,t88_xcmplx_1]), [file(borsuk_5,t44_borsuk_5),interesting(0.67)]). fof(l87_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => r1_tarski(k12_prob_1(A),k2_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[t15_limfunc1]), [file(borsuk_5,l87_borsuk_5),interesting(0.66)]). fof(t26_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => r1_xboole_0(k2_limfunc1(A),k4_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[l26_borsuk_5,t79_xboole_1]), [file(borsuk_5,t26_borsuk_5),interesting(0.66)]). fof(l86_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => r1_tarski(k4_limfunc1(A),k3_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[t10_limfunc1]), [file(borsuk_5,l86_borsuk_5),interesting(0.66)]). fof(t3_borsuk_5,theorem,( ! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k1_enumset1(A,C,F),k1_enumset1(B,D,E)) ), inference(mizar_proof,[status(thm)],[d3_tarski,d4_enumset1,d1_enumset1,d2_xboole_0,d10_xboole_0,d3_tarski,d2_xboole_0,d1_enumset1,d4_enumset1]), [file(borsuk_5,t3_borsuk_5),interesting(0.66)]). fof(t7_borsuk_5,theorem,( ! [A,B,C,D,E,F] : ( ( r1_incproj(A,B,C) & r1_incproj(D,E,F) & r1_xboole_0(k1_enumset1(A,B,C),k1_enumset1(D,E,F)) ) => r1_borsuk_5(A,B,C,D,E,F) ) ), inference(mizar_proof,[status(thm)],[d5_incproj,d5_incproj,t6_borsuk_5,d1_borsuk_5]), [file(borsuk_5,t7_borsuk_5),interesting(0.66)]). fof(t2_borsuk_5,theorem,( ! [A,B,C] : ~ ( r1_tarski(A,B) & r1_tarski(B,k2_xboole_0(A,k1_tarski(C))) & k2_xboole_0(A,k1_tarski(C)) != B & A != B ) ), inference(mizar_proof,[status(thm)],[d10_xboole_0,t37_zfmisc_1,t8_xboole_1,t5_setwiseo]), [file(borsuk_5,t2_borsuk_5),interesting(0.65)]). fof(t13_borsuk_5,theorem,( k2_pre_topc(k3_topmetr) = k1_numbers ), inference(mizar_proof,[status(thm)],[d3_pre_topc,t24_topmetr]), [file(borsuk_5,t13_borsuk_5),interesting(0.65)]). fof(t14_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r2_hidden(A,k4_limfunc1(B)) <=> ~ r1_xreal_0(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d3_limfunc1,d3_limfunc1]), [file(borsuk_5,t14_borsuk_5),interesting(0.65)]). fof(t15_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r2_hidden(A,k3_limfunc1(B)) <=> r1_xreal_0(B,A) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d2_limfunc1,d2_limfunc1]), [file(borsuk_5,t15_borsuk_5),interesting(0.65)]). fof(l113_borsuk_5,theorem,( r1_tarski(k4_limfunc1(1),k3_limfunc1(1)) ), inference(mizar_proof,[status(thm)],[l86_borsuk_5]), [file(borsuk_5,l113_borsuk_5),interesting(0.65)]). fof(t16_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r2_hidden(A,k2_limfunc1(B)) <=> r1_xreal_0(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d1_limfunc1,d1_limfunc1]), [file(borsuk_5,t16_borsuk_5),interesting(0.63)]). fof(t47_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & ! [C] : ( v1_rat_1(C) => ! [D] : ( v1_rat_1(D) => ~ ( ~ r1_xreal_0(C,A) & ~ r1_xreal_0(D,C) & ~ r1_xreal_0(B,D) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t22_rat_1,t22_rat_1]), [file(borsuk_5,t47_borsuk_5),interesting(0.63)]). fof(t56_borsuk_5,theorem,( ! [A] : ( ( v2_pre_topc(A) & v1_connsp_1(A) & l1_pre_topc(A) ) => ! [B] : ( ( v3_pre_topc(B,A) & v4_pre_topc(B,A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( B = k1_xboole_0 | B = k2_pre_topc(A) ) ) ) ), inference(mizar_proof,[status(thm)],[t18_pre_topc,t20_tops_1,t3_connsp_1,t23_pre_topc,t17_pre_topc,t16_connsp_1,t28_connsp_1]), [file(borsuk_5,t56_borsuk_5),interesting(0.62)]). fof(t57_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ~ ( v4_pre_topc(A,k3_topmetr) & v3_pre_topc(A,k3_topmetr) & A != k1_xboole_0 & A != k1_numbers ) ) ), inference(mizar_proof,[status(thm)],[t56_borsuk_5,d3_pre_topc,t24_topmetr]), [file(borsuk_5,t57_borsuk_5),interesting(0.62)]). fof(t53_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(B,A) => ( r2_hidden(C,k3_borsuk_5(A,B)) <=> ( ~ v1_rat_1(C) & ~ r1_xreal_0(C,A) & ~ r1_xreal_0(B,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_xboole_0,t37_borsuk_5,t47_rcomp_1,t37_borsuk_5,t47_rcomp_1,d3_xboole_0]), [file(borsuk_5,t53_borsuk_5),interesting(0.62)]). fof(l104_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(B,A) => k2_borsuk_5(A,B) = k1_xboole_0 ) ) ) ), inference(mizar_proof,[status(thm)],[t12_rcomp_1]), [file(borsuk_5,l104_borsuk_5),interesting(0.62)]). fof(t17_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r2_hidden(A,k12_prob_1(B)) <=> ~ r1_xreal_0(B,A) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d15_prob_1,d15_prob_1]), [file(borsuk_5,t17_borsuk_5),interesting(0.61)]). fof(t29_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,C) & r1_xreal_0(C,B) ) => k4_subset_1(k1_numbers,k2_limfunc1(C),k1_rcomp_1(A,B)) = k2_limfunc1(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,t16_borsuk_5,t2_xreal_1,t16_borsuk_5,t48_rcomp_1,t16_borsuk_5,d2_xboole_0,d10_xboole_0,d3_tarski,d1_xreal_0,t16_borsuk_5,d2_xboole_0,t2_xreal_1,t16_borsuk_5,t48_rcomp_1,d2_xboole_0]), [file(borsuk_5,t29_borsuk_5),interesting(0.61)]). fof(t45_borsuk_5,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v1_rat_1(A) ) => ~ v1_rat_1(k4_int_1(A)) ) ), inference(mizar_proof,[status(thm)],[d6_int_1,t41_borsuk_5]), [file(borsuk_5,t45_borsuk_5),interesting(0.61)]). fof(l105_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(B,A) => k1_numbers = k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t2_xreal_1,t16_borsuk_5,d2_xboole_0,t15_borsuk_5,d2_xboole_0,d10_xboole_0]), [file(borsuk_5,l105_borsuk_5),interesting(0.61)]). fof(t6_borsuk_5,theorem,( ! [A,B,C,D,E,F] : ( r1_xboole_0(k1_enumset1(A,B,C),k1_enumset1(D,E,F)) => ( A != D & A != E & A != F & B != D & B != E & B != F & C != D & C != E & C != F ) ) ), inference(mizar_proof,[status(thm)],[d1_enumset1,d1_enumset1,t3_xboole_0]), [file(borsuk_5,t6_borsuk_5),interesting(0.60)]). fof(t28_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,C) & r1_xreal_0(C,B) ) => k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k3_limfunc1(C)) = k3_limfunc1(A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t15_borsuk_5,t48_rcomp_1,d2_xboole_0,t2_xreal_1,t15_borsuk_5,d2_xboole_0,t12_limfunc1,t9_limfunc1,t8_xboole_1,d10_xboole_0]), [file(borsuk_5,t28_borsuk_5),interesting(0.60)]). fof(t35_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = k3_numbers => k6_pre_topc(k3_topmetr,A) = u1_struct_0(k3_topmetr) ) ) ), inference(mizar_proof,[status(thm)],[d10_xboole_0,d3_tarski,d14_metric_1,t24_topmetr,d14_metric_1,d1_xreal_0,t31_xreal_1,t22_rat_1,d2_rat_1,d14_metric_1,d1_xreal_0,t11_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr]), [file(borsuk_5,t35_borsuk_5),interesting(0.60)]). fof(t94_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k6_subset_1(k1_numbers,k4_limfunc1(A),k1_seq_4(B)) = k4_subset_1(k1_numbers,k2_rcomp_1(A,B),k4_limfunc1(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,t64_zfmisc_1,t14_borsuk_5,t14_borsuk_5,d2_xboole_0,t47_rcomp_1,d2_xboole_0,d5_real_1,d10_xboole_0,d3_tarski,d1_xreal_0,t14_borsuk_5,t2_xreal_1,t14_borsuk_5,t64_zfmisc_1,t47_rcomp_1,t14_borsuk_5,t64_zfmisc_1,d2_xboole_0]), [file(borsuk_5,t94_borsuk_5),interesting(0.59)]). fof(t52_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(B,A) => ( r2_hidden(C,k2_borsuk_5(A,B)) <=> ( v1_rat_1(C) & ~ r1_xreal_0(C,A) & ~ r1_xreal_0(B,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_xboole_0,t47_rcomp_1,d2_rat_1,t47_rcomp_1,d2_rat_1,d3_xboole_0]), [file(borsuk_5,t52_borsuk_5),interesting(0.59)]). fof(t93_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k6_subset_1(k1_numbers,k12_prob_1(B),k1_seq_4(A)) = k4_subset_1(k1_numbers,k12_prob_1(A),k2_rcomp_1(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,t64_zfmisc_1,t17_borsuk_5,t17_borsuk_5,d2_xboole_0,t47_rcomp_1,d2_xboole_0,d5_real_1,d10_xboole_0,d3_tarski,d1_xreal_0,t17_borsuk_5,t2_xreal_1,t17_borsuk_5,t64_zfmisc_1,t47_rcomp_1,t17_borsuk_5,t64_zfmisc_1,d2_xboole_0]), [file(borsuk_5,t93_borsuk_5),interesting(0.59)]). fof(t67_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k3_limfunc1(A) = k4_subset_1(k1_numbers,k1_seq_4(A),k4_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,t15_borsuk_5,d1_tarski,d2_xboole_0,t14_borsuk_5,d2_xboole_0,d5_real_1,d10_xboole_0,d3_tarski,d1_xreal_0,d2_xboole_0,t14_borsuk_5,d1_tarski,t15_borsuk_5]), [file(borsuk_5,t67_borsuk_5),interesting(0.58)]). fof(l63_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_borsuk_5(B,C) => ( r1_xreal_0(C,B) | ( r2_hidden(B,k6_pre_topc(k3_topmetr,A)) & r2_hidden(C,k6_pre_topc(k3_topmetr,A)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d14_metric_1,d1_xreal_0,d14_metric_1,d1_xreal_0,t35_square_1,t3_topreal3,t31_xreal_1,t3_topreal3,t38_square_1,t22_rat_1,d2_rat_1,t2_xreal_1,t2_xreal_1,t47_rcomp_1,d3_xboole_0,d14_metric_1,d1_xreal_0,t11_xreal_1,t35_square_1,t11_xreal_1,t2_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr,d14_metric_1,d1_xreal_0,d14_metric_1,d1_xreal_0,t31_xreal_1,t46_square_1,t3_topreal3,t21_xreal_1,t3_topreal3,t49_square_1,t22_rat_1,d2_rat_1,t2_xreal_1,t2_xreal_1,t47_rcomp_1,d3_xboole_0,d14_metric_1,d1_xreal_0,t105_real_2,t46_square_1,t92_real_1,t2_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr]), [file(borsuk_5,l63_borsuk_5),interesting(0.58)]). fof(t27_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => r1_xboole_0(k12_prob_1(A),k3_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[l25_borsuk_5,t79_xboole_1]), [file(borsuk_5,t27_borsuk_5),interesting(0.57)]). fof(t40_borsuk_5,theorem,( ! [A] : ( v1_rat_1(A) => ! [B] : ( ( v1_xreal_0(B) & ~ v1_rat_1(B) ) => ~ v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[t39_borsuk_5,t38_borsuk_5]), [file(borsuk_5,t40_borsuk_5),interesting(0.57)]). fof(t64_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k12_prob_1(B) => v3_pre_topc(A,k3_topmetr) ) ) ) ), inference(mizar_proof,[status(thm)],[l78_borsuk_5,t62_borsuk_5]), [file(borsuk_5,t64_borsuk_5),interesting(0.56)]). fof(t34_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k8_metric_1)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k8_metric_1)) => ( r1_xreal_0(A,B) => k4_metric_1(k8_metric_1,A,B) = k6_xcmplx_0(B,A) ) ) ) ), inference(mizar_proof,[status(thm)],[t50_xreal_1,t15_topmetr,d1_absvalue]), [file(borsuk_5,t34_borsuk_5),interesting(0.56)]). fof(t63_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k4_limfunc1(B) => v3_pre_topc(A,k3_topmetr) ) ) ) ), inference(mizar_proof,[status(thm)],[l76_borsuk_5,t62_borsuk_5]), [file(borsuk_5,t63_borsuk_5),interesting(0.56)]). fof(t65_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k2_limfunc1(B) => v4_pre_topc(A,k3_topmetr) ) ) ) ), inference(mizar_proof,[status(thm)],[l77_borsuk_5,t79_topreal6]), [file(borsuk_5,t65_borsuk_5),interesting(0.54)]). fof(t66_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k3_limfunc1(B) => v4_pre_topc(A,k3_topmetr) ) ) ) ), inference(mizar_proof,[status(thm)],[l79_borsuk_5,t79_topreal6]), [file(borsuk_5,t66_borsuk_5),interesting(0.54)]). fof(t73_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k2_limfunc1(A) != k1_numbers ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t31_xreal_1,t16_borsuk_5]), [file(borsuk_5,t73_borsuk_5),interesting(0.53)]). fof(t80_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k4_subset_1(k1_numbers,k1_rcomp_2(B,C),k4_limfunc1(C)) => ( r1_xreal_0(C,B) | k6_pre_topc(k3_topmetr,A) = k3_limfunc1(B) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t58_borsuk_5,t75_borsuk_5,t50_pre_topc,t28_borsuk_5]), [file(borsuk_5,t80_borsuk_5),interesting(0.53)]). fof(t96_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k4_subset_1(k1_numbers,k12_prob_1(B),k2_rcomp_1(B,C)) => ( r1_xreal_0(C,B) | k6_pre_topc(k3_topmetr,A) = k2_limfunc1(C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t36_borsuk_5,t50_pre_topc,t77_borsuk_5,t29_borsuk_5]), [file(borsuk_5,t96_borsuk_5),interesting(0.53)]). fof(t62_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = B => ( v3_rcomp_1(A) <=> v3_pre_topc(B,k3_topmetr) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t6_jordan5a,t16_topmetr,d5_pre_topc,d7_topmetr,d5_pre_topc,t16_topmetr,d7_topmetr,t6_jordan5a]), [file(borsuk_5,t62_borsuk_5),interesting(0.52)]). fof(t72_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k3_limfunc1(A) != k1_numbers ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t148_xreal_1,t15_borsuk_5]), [file(borsuk_5,t72_borsuk_5),interesting(0.52)]). fof(t82_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( A = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_borsuk_5(B,C),k2_rcomp_1(C,D)),k4_limfunc1(D)) => ( r1_xreal_0(C,B) | r1_xreal_0(D,C) | k6_pre_topc(k3_topmetr,A) = k3_limfunc1(B) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t24_topmetr,t4_xboole_1,t50_pre_topc,t81_borsuk_5,t54_borsuk_5,t28_borsuk_5]), [file(borsuk_5,t82_borsuk_5),interesting(0.51)]). fof(t61_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k1_seq_4(B) => k6_pre_topc(k3_topmetr,A) = k1_seq_4(B) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,d1_xreal_0,t35_yellow_8]), [file(borsuk_5,t61_borsuk_5),interesting(0.50)]). fof(t55_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k3_borsuk_5(B,C) => ( r1_xreal_0(C,B) | k6_pre_topc(k3_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t24_topmetr,t51_pre_topc,t28_xboole_1,d3_tarski,t51_borsuk_5,t36_borsuk_5,d10_xboole_0,d3_tarski,d14_metric_1,t48_rcomp_1,t48_rcomp_1,d14_metric_1,d1_xreal_0,t35_square_1,t3_topreal3,t31_xreal_1,t3_topreal3,t38_square_1,t50_borsuk_5,t37_borsuk_5,t2_xreal_1,t2_xreal_1,t47_rcomp_1,d3_xboole_0,d14_metric_1,d1_xreal_0,t11_xreal_1,t35_square_1,t11_xreal_1,t2_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr,l64_borsuk_5,d5_real_1]), [file(borsuk_5,t55_borsuk_5),interesting(0.50)]). fof(t19_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_xreal_0(B,C) => ( r1_xreal_0(B,A) | r1_xboole_0(k1_rcomp_1(A,B),k2_rcomp_2(C,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,d1_xreal_0,t4_rcomp_2,t48_rcomp_1,t2_xreal_1]), [file(borsuk_5,t19_borsuk_5),interesting(0.48)]). fof(t20_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_xreal_0(B,C) => ( r1_xreal_0(B,A) | r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_1(C,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,d1_xreal_0,t3_rcomp_2,t48_rcomp_1,t2_xreal_1]), [file(borsuk_5,t20_borsuk_5),interesting(0.48)]). fof(t59_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_rcomp_2(B,C) => ( B = C | k6_pre_topc(k3_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t20_borsuk_4,t80_topreal6]), [file(borsuk_5,t59_borsuk_5),interesting(0.47)]). fof(t36_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_rcomp_1(B,C) => ( B = C | k6_pre_topc(k3_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t82_topreal6,t80_topreal6]), [file(borsuk_5,t36_borsuk_5),interesting(0.47)]). fof(t104_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k4_subset_1(k1_numbers,k2_limfunc1(A),k1_seq_4(B)) != k1_numbers ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t46_square_1,t31_xreal_1,t2_xreal_1,t16_borsuk_5,d1_tarski,d2_xboole_0]), [file(borsuk_5,t104_borsuk_5),interesting(0.47)]). fof(t58_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k1_rcomp_2(B,C) => ( B = C | k6_pre_topc(k3_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t21_borsuk_4,t80_topreal6]), [file(borsuk_5,t58_borsuk_5),interesting(0.47)]). fof(t103_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k4_subset_1(k1_numbers,k3_limfunc1(A),k1_seq_4(B)) != k1_numbers ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t35_square_1,t148_xreal_1,t2_xreal_1,t15_borsuk_5,d1_tarski,d2_xboole_0]), [file(borsuk_5,t103_borsuk_5),interesting(0.46)]). fof(t18_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => k4_xboole_0(k1_numbers,k1_seq_4(A)) = k4_subset_1(k1_numbers,k12_prob_1(A),k4_limfunc1(A)) ) ), inference(mizar_proof,[status(thm)],[t14_rcomp_1,t25_limfunc1]), [file(borsuk_5,t18_borsuk_5),interesting(0.46)]). fof(t43_borsuk_5,theorem,( ! [A] : ( v1_rat_1(A) => ! [B] : ( ( v1_xreal_0(B) & ~ v1_rat_1(B) ) => ~ ( A != 0 & v1_rat_1(k7_xcmplx_0(B,A)) ) ) ) ), inference(mizar_proof,[status(thm)],[t88_xcmplx_1]), [file(borsuk_5,t43_borsuk_5),interesting(0.43)]). fof(l112_borsuk_5,theorem,( r1_tarski(k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k3_borsuk_5(2,4),k1_seq_4(4)),k1_seq_4(5)),k4_limfunc1(1)) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,d2_xboole_0,t53_borsuk_5,t2_xreal_1,t14_borsuk_5,d1_tarski,t14_borsuk_5,d1_tarski,t14_borsuk_5,d2_xboole_0]), [file(borsuk_5,l112_borsuk_5),interesting(0.43)]). fof(t100_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( A = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k12_prob_1(B),k2_rcomp_2(B,C)),k3_borsuk_5(C,D)),k1_seq_4(D)) => ( r1_xreal_0(C,B) | r1_xreal_0(D,C) | k6_pre_topc(k3_topmetr,A) = k2_limfunc1(D) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t16_borsuk_5,d1_tarski,t50_pre_topc,t61_borsuk_5,t50_pre_topc,t97_borsuk_5,t55_borsuk_5,t29_borsuk_5,t46_zfmisc_1]), [file(borsuk_5,t100_borsuk_5),interesting(0.42)]). fof(t87_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & r2_hidden(B,k4_subset_1(k1_numbers,k2_rcomp_1(A,B),k4_limfunc1(B))) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_borsuk_5,t47_rcomp_1,d2_xboole_0]), [file(borsuk_5,t87_borsuk_5),interesting(0.42)]). fof(t91_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => k5_subset_1(k1_numbers,k4_limfunc1(A),k2_limfunc1(B)) = k2_rcomp_2(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,d3_xboole_0,t14_borsuk_5,t16_borsuk_5,t4_rcomp_2,d10_xboole_0,d3_tarski,d1_xreal_0,t4_rcomp_2,t14_borsuk_5,t16_borsuk_5,d3_xboole_0]), [file(borsuk_5,t91_borsuk_5),interesting(0.42)]). fof(t107_borsuk_5,theorem,( ! [A] : ( ( v6_compts_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_numbers)) => ( B = A => ( v1_seq_4(B) & v2_seq_4(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t81_topreal6,t28_rcomp_1,d3_seq_4]), [file(borsuk_5,t107_borsuk_5),interesting(0.40)]). fof(t86_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( r1_xreal_0(A,B) & ~ r1_xreal_0(C,B) & r2_hidden(A,k4_subset_1(k1_numbers,k2_rcomp_1(B,C),k4_limfunc1(C))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_xreal_1,t14_borsuk_5,t47_rcomp_1,d2_xboole_0]), [file(borsuk_5,t86_borsuk_5),interesting(0.40)]). fof(l114_borsuk_5,theorem,( r1_tarski(k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k3_borsuk_5(2,4),k1_seq_4(4)),k1_seq_4(5)),k3_limfunc1(1)) ), inference(mizar_proof,[status(thm)],[l113_borsuk_5,l112_borsuk_5,t1_xboole_1]), [file(borsuk_5,l114_borsuk_5),interesting(0.36)]). fof(t109_borsuk_5,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k3_topmetr) & v6_compts_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_numbers)) => ( ( A = B & r1_tarski(k1_rcomp_1(k4_pscomp_1(B),k3_pscomp_1(B)),B) ) => k1_rcomp_1(k4_pscomp_1(B),k3_pscomp_1(B)) = B ) ) ) ), inference(mizar_proof,[status(thm)],[d10_xboole_0,d3_tarski,d1_xreal_0,t108_borsuk_5,t48_rcomp_1]), [file(borsuk_5,t109_borsuk_5),interesting(0.35)]). fof(t83_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k4_xboole_0(k1_numbers,A) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc]), [file(borsuk_5,t83_borsuk_5),interesting(0.35)]). fof(t98_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k2_limfunc1(B) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k4_limfunc1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc,l26_borsuk_5]), [file(borsuk_5,t98_borsuk_5),interesting(0.33)]). fof(t99_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k3_limfunc1(B) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k12_prob_1(B) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc,l27_borsuk_5]), [file(borsuk_5,t99_borsuk_5),interesting(0.33)]). fof(l64_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k3_borsuk_5(B,C) => ( r1_xreal_0(C,B) | ( r2_hidden(B,k6_pre_topc(k3_topmetr,A)) & r2_hidden(C,k6_pre_topc(k3_topmetr,A)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d14_metric_1,d1_xreal_0,d14_metric_1,d1_xreal_0,t35_square_1,t3_topreal3,t31_xreal_1,t3_topreal3,t38_square_1,t50_borsuk_5,t37_borsuk_5,t2_xreal_1,t2_xreal_1,t47_rcomp_1,d3_xboole_0,d14_metric_1,d1_xreal_0,t11_xreal_1,t35_square_1,t11_xreal_1,t2_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr,d14_metric_1,d1_xreal_0,d14_metric_1,d1_xreal_0,t31_xreal_1,t46_square_1,t3_topreal3,t21_xreal_1,t3_topreal3,t49_square_1,t50_borsuk_5,t37_borsuk_5,t2_xreal_1,t2_xreal_1,t47_rcomp_1,d3_xboole_0,d14_metric_1,d1_xreal_0,t105_real_2,t46_square_1,t92_real_1,t2_xreal_1,t34_borsuk_5,t12_metric_1,t3_xboole_0,t7_hausdorf,d7_topmetr]), [file(borsuk_5,l64_borsuk_5),interesting(0.32)]). fof(t106_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( k1_numbers = k3_subset_1(u1_struct_0(k3_topmetr),A) => A = k1_xboole_0 ) ) ), inference(mizar_proof,[status(thm)],[t12_pre_topc,t24_topmetr,t27_pre_topc]), [file(borsuk_5,t106_borsuk_5),interesting(0.32)]). fof(t114_borsuk_5,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( ( v1_borsuk_5(B,A) & r1_tarski(C,B) ) => v1_borsuk_5(C,A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d6_borsuk_5,d6_borsuk_5]), [file(borsuk_5,t114_borsuk_5),interesting(0.32)]). fof(t60_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( A = k4_subset_1(k1_numbers,k1_rcomp_2(B,C),k2_rcomp_2(C,D)) => ( r1_xreal_0(C,B) | r1_xreal_0(D,C) | k6_pre_topc(k3_topmetr,A) = k1_rcomp_1(B,D) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t50_pre_topc,t58_borsuk_5,t59_borsuk_5,t2_treal_1]), [file(borsuk_5,t60_borsuk_5),interesting(0.21)]). fof(t89_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_borsuk_5(2,4),k2_rcomp_1(4,5)),k4_limfunc1(5)) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_limfunc1(2),k3_borsuk_5(2,4)),k1_seq_4(4)),k1_seq_4(5)) ) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc,t4_xboole_1,t41_xboole_1,t85_borsuk_5,t26_borsuk_5,d3_tarski,d1_xreal_0,t16_borsuk_5,t2_xreal_1,t16_borsuk_5,t53_borsuk_5,t16_borsuk_5,d2_xboole_0,d3_tarski,d1_xreal_0,t47_rcomp_1,t14_borsuk_5,t14_borsuk_5,t2_xreal_1,t14_borsuk_5,d2_xboole_0,t64_xboole_1,t87_xboole_1,t88_borsuk_5,t4_xboole_1]), [file(borsuk_5,t89_borsuk_5),interesting(0.21)]). fof(t102_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(B,C) & A = k4_subset_1(k1_numbers,k2_limfunc1(B),k1_seq_4(C)) ) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k4_subset_1(k1_numbers,k2_rcomp_1(B,C),k4_limfunc1(C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc,t41_xboole_1,l26_borsuk_5,t94_borsuk_5]), [file(borsuk_5,t102_borsuk_5),interesting(0.19)]). fof(t95_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(B,C) & A = k4_subset_1(k1_numbers,k1_seq_4(B),k3_limfunc1(C)) ) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k4_subset_1(k1_numbers,k12_prob_1(B),k2_rcomp_1(B,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc,t41_xboole_1,l27_borsuk_5,t93_borsuk_5]), [file(borsuk_5,t95_borsuk_5),interesting(0.19)]). fof(t11_borsuk_5,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] : ( 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 ) & ! [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) = C & k1_funct_1(D,1) = B ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_treal_1,t12_treal_1,d2_treal_1,d2_treal_1,t12_treal_1,d1_treal_1,t19_funct_2,t27_topmetr,t48_rcomp_1,t25_topmetr,d1_funct_2,t48_rcomp_1,t25_topmetr,d1_funct_2,t11_treal_1,t27_topmetr,t58_tops_2,t23_funct_1]), [file(borsuk_5,t11_borsuk_5),interesting(0.14)]). fof(t101_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ! [E] : ( v1_xreal_0(E) => ( A = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k12_prob_1(B),k2_rcomp_2(B,C)),k3_borsuk_5(C,D)),k1_seq_4(D)),k1_seq_4(E)) => ( r1_xreal_0(C,B) | r1_xreal_0(D,C) | k6_pre_topc(k3_topmetr,A) = k4_subset_1(k1_numbers,k2_limfunc1(D),k1_seq_4(E)) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_topmetr,t50_pre_topc,t61_borsuk_5,t100_borsuk_5]), [file(borsuk_5,t101_borsuk_5),interesting(0.07)]). 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(t24_topmetr,theorem,( u1_struct_0(k3_topmetr) = k1_numbers ), file(topmetr,t24_topmetr), [interesting(0.00)]). fof(t48_rcomp_1,theorem,( ! [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), [interesting(0.00)]). fof(t26_topmetr,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( C = k1_rcomp_1(A,B) => k4_topmetr(A,B) = k3_pre_topc(k3_topmetr,C) ) ) ) ) ) ), file(topmetr,t26_topmetr), [interesting(0.00)]). fof(t27_topmetr,theorem,( k4_topmetr(0,1) = k22_borsuk_1 ), file(topmetr,t27_topmetr), [interesting(0.00)]). fof(t11_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B))) => ( v1_funct_1(k3_treal_1(A,B,C,D)) & v1_funct_2(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(0,1)),u1_struct_0(k4_topmetr(A,B))) & v5_pre_topc(k3_treal_1(A,B,C,D),k4_topmetr(0,1),k4_topmetr(A,B)) & m2_relset_1(k3_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(0,1)),u1_struct_0(k4_topmetr(A,B))) ) ) ) ) ) ) ), file(treal_1,t11_treal_1), [interesting(0.00)]). fof(t35_borsuk_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => r1_tarski(u1_struct_0(B),u1_struct_0(A)) ) ) ), file(borsuk_1,t35_borsuk_1), [interesting(0.00)]). fof(t9_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r1_tarski(B,C) => ( ( B = k1_xboole_0 & A != k1_xboole_0 ) | ( v1_funct_1(D) & v1_funct_2(D,A,C) & m2_relset_1(D,A,C) ) ) ) ) ), file(funct_2,t9_funct_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(t7_topmetr,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v2_pre_topc(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)) ) => ( ( v5_pre_topc(D,A,C) & m1_pre_topc(C,B) ) => ! [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)) ) => ( E = D => v5_pre_topc(E,A,B) ) ) ) ) ) ) ) ), file(topmetr,t7_topmetr), [interesting(0.00)]). fof(d1_treal_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k1_treal_1(A,B) = A ) ) ) ), file(treal_1,d1_treal_1), [interesting(0.00)]). fof(t12_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ! [C] : ( m1_subset_1(C,u1_struct_0(k4_topmetr(A,B))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k4_topmetr(A,B))) => ( k8_funct_2(u1_struct_0(k4_topmetr(0,1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),k1_treal_1(0,1)) = C & k8_funct_2(u1_struct_0(k4_topmetr(0,1)),u1_struct_0(k4_topmetr(A,B)),k3_treal_1(A,B,C,D),k2_treal_1(0,1)) = D ) ) ) ) ) ) ), file(treal_1,t12_treal_1), [interesting(0.00)]). fof(d2_treal_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k2_treal_1(A,B) = B ) ) ) ), file(treal_1,d2_treal_1), [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(t19_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,B,C) & m2_relset_1(E,B,C) ) => ( ~ ( B = k1_xboole_0 & C != k1_xboole_0 & A != k1_xboole_0 ) => ( v1_funct_1(k1_partfun1(A,B,B,C,D,E)) & v1_funct_2(k1_partfun1(A,B,B,C,D,E),A,C) & m2_relset_1(k1_partfun1(A,B,B,C,D,E),A,C) ) ) ) ) ), file(funct_2,t19_funct_2), [interesting(0.00)]). fof(t25_topmetr,theorem,( ! [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), [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(t58_tops_2,theorem,( ! [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), [interesting(0.00)]). fof(t23_funct_1,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(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ), file(funct_1,t23_funct_1), [interesting(0.00)]). fof(t50_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(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))) => k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) ), file(pre_topc,t50_pre_topc), [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(t35_yellow_8,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_urysohn1(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B) ) ) ) ), file(yellow_8,t35_yellow_8), [interesting(0.00)]). fof(d1_limfunc1,definition,( ! [A] : ( v1_xreal_0(A) => k2_limfunc1(A) = a_1_0_limfunc1(A) ) ), file(limfunc1,d1_limfunc1), [interesting(0.00)]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.00)]). fof(t20_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( A != B => k7_pscomp_1(k2_rcomp_2(A,B)) = k1_rcomp_1(A,B) ) ) ) ), file(borsuk_4,t20_borsuk_4), [interesting(0.00)]). fof(t80_topreal6,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = B => k7_pscomp_1(A) = k6_pre_topc(k3_topmetr,B) ) ) ) ), file(topreal6,t80_topreal6), [interesting(0.00)]). fof(t8_fcont_3,theorem,( ! [A] : ( v1_xreal_0(A) => v3_rcomp_1(k12_prob_1(A)) ) ), file(fcont_3,t8_fcont_3), [interesting(0.00)]). fof(t6_jordan5a,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( r2_hidden(A,k4_pcomps_1(k8_metric_1)) <=> v3_rcomp_1(A) ) ) ), file(jordan5a,t6_jordan5a), [interesting(0.00)]). fof(t16_topmetr,theorem,( ! [A] : ( l1_metric_1(A) => ( u1_struct_0(A) = u1_struct_0(k5_pcomps_1(A)) & u1_pre_topc(k5_pcomps_1(A)) = k4_pcomps_1(A) ) ) ), file(topmetr,t16_topmetr), [interesting(0.00)]). fof(d5_pre_topc,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v3_pre_topc(B,A) <=> r2_hidden(B,u1_pre_topc(A)) ) ) ) ), file(pre_topc,d5_pre_topc), [interesting(0.00)]). fof(d7_topmetr,definition,( k3_topmetr = k5_pcomps_1(k8_metric_1) ), file(topmetr,d7_topmetr), [interesting(0.00)]). fof(t6_fcont_3,theorem,( ! [A] : ( v1_xreal_0(A) => v2_rcomp_1(k2_limfunc1(A)) ) ), file(fcont_3,t6_fcont_3), [interesting(0.00)]). fof(t79_topreal6,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = B => ( v2_rcomp_1(A) <=> v4_pre_topc(B,k3_topmetr) ) ) ) ) ), file(topreal6,t79_topreal6), [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(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(d15_prob_1,definition,( ! [A] : ( v1_xreal_0(A) => k12_prob_1(A) = a_1_0_prob_1(A) ) ), file(prob_1,d15_prob_1), [interesting(0.00)]). fof(d5_real_1,definition,( ! [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), [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(t48_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => r1_tarski(B,k6_pre_topc(A,B)) ) ) ), file(pre_topc,t48_pre_topc), [interesting(0.00)]). fof(t15_limfunc1,theorem,( ! [A] : ( v1_xreal_0(A) => r1_tarski(k12_prob_1(A),k2_limfunc1(A)) ) ), file(limfunc1,t15_limfunc1), [interesting(0.00)]). fof(t31_tops_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))) => ( ( v4_pre_topc(B,A) & r1_tarski(C,B) ) => r1_tarski(k6_pre_topc(A,C),B) ) ) ) ) ), file(tops_1,t31_tops_1), [interesting(0.00)]). fof(t18_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => k4_subset_1(u1_struct_0(A),B,k3_subset_1(u1_struct_0(A),B)) = k2_pre_topc(A) ) ) ), file(pre_topc,t18_pre_topc), [interesting(0.00)]). fof(t20_tops_1,theorem,( ! [A] : ( l1_struct_0(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))) => ( r1_tarski(B,C) <=> r1_xboole_0(B,k3_subset_1(u1_struct_0(A),C)) ) ) ) ) ), file(tops_1,t20_tops_1), [interesting(0.00)]). fof(t3_connsp_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))) => ( ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C) & v4_pre_topc(B,A) & v4_pre_topc(C,A) & r1_xboole_0(B,C) ) => r1_connsp_1(A,B,C) ) ) ) ) ), file(connsp_1,t3_connsp_1), [interesting(0.00)]). fof(t23_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ~ ( B != k2_pre_topc(A) & k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) = k1_xboole_0 ) & ~ ( k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) != k1_xboole_0 & B = k2_pre_topc(A) ) ) ) ) ), file(pre_topc,t23_pre_topc), [interesting(0.00)]). fof(t17_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => k3_subset_1(u1_struct_0(A),B) = k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B) ) ) ), file(pre_topc,t17_pre_topc), [interesting(0.00)]). fof(t16_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_connsp_1(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(A))) => ~ ( B = k4_subset_1(u1_struct_0(A),C,D) & r1_connsp_1(A,C,D) & C != k1_pre_topc(A) & D != k1_pre_topc(A) ) ) ) ) ) ) ), file(connsp_1,t16_connsp_1), [interesting(0.00)]). fof(t28_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v2_connsp_1(k2_pre_topc(A),A) <=> v1_connsp_1(A) ) ) ), file(connsp_1,t28_connsp_1), [interesting(0.00)]). fof(d3_pre_topc,definition,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,d3_pre_topc), [interesting(0.00)]). fof(t31_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), [interesting(0.00)]). fof(t37_zfmisc_1,theorem,( ! [A,B] : ( r1_tarski(k1_tarski(A),B) <=> r2_hidden(A,B) ) ), file(zfmisc_1,t37_zfmisc_1), [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(t5_setwiseo,theorem,( ! [A,B,C] : ~ ( r1_tarski(A,k2_xboole_0(B,k1_tarski(C))) & ~ r2_hidden(C,A) & ~ r1_tarski(A,B) ) ), file(setwiseo,t5_setwiseo), [interesting(0.00)]). fof(t2_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(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.00)]). fof(t51_pre_topc,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))) => r1_tarski(k6_pre_topc(A,k5_subset_1(u1_struct_0(A),B,C)),k5_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C))) ) ) ) ), file(pre_topc,t51_pre_topc), [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(d14_metric_1,definition,( k8_metric_1 = g1_metric_1(k1_numbers,k7_metric_1) ), file(metric_1,d14_metric_1), [interesting(0.00)]). fof(t22_rat_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & ! [C] : ( v1_rat_1(C) => ~ ( ~ r1_xreal_0(C,A) & ~ r1_xreal_0(B,C) ) ) ) ) ) ), file(rat_1,t22_rat_1), [interesting(0.00)]). fof(t3_rat_1,theorem,( ! [A] : ~ ( v1_rat_1(A) & ! [B] : ( v1_int_1(B) => ! [C] : ( v1_int_1(C) => ~ ( C != 0 & A = k7_xcmplx_0(B,C) ) ) ) ) ), file(rat_1,t3_rat_1), [interesting(0.00)]). fof(t85_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => ! [D] : ( v1_xcmplx_0(D) => k7_xcmplx_0(k7_xcmplx_0(A,B),k7_xcmplx_0(C,D)) = k7_xcmplx_0(k3_xcmplx_0(A,D),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xcmplx_1,t85_xcmplx_1), [interesting(0.00)]). fof(t90_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => B = k7_xcmplx_0(k3_xcmplx_0(B,A),A) ) ) ) ), file(xcmplx_1,t90_xcmplx_1), [interesting(0.00)]). fof(t131_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => ! [D] : ( v1_xcmplx_0(D) => ~ ( A != 0 & B != 0 & k6_xcmplx_0(k7_xcmplx_0(C,A),k7_xcmplx_0(D,B)) != k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(C,B),k3_xcmplx_0(D,A)),k3_xcmplx_0(A,B)) ) ) ) ) ) ), file(xcmplx_1,t131_xcmplx_1), [interesting(0.00)]). fof(t69_int_1,theorem,( ! [A] : ( v1_xreal_0(A) => ( ~ r1_xreal_0(1,k4_int_1(A)) & r1_xreal_0(0,k4_int_1(A)) ) ) ), file(int_1,t69_int_1), [interesting(0.00)]). fof(t180_xreal_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,0) & ~ r1_xreal_0(1,A) & ~ r1_xreal_0(C,B) & r1_xreal_0(D,C) & r1_xreal_0(C,k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(1,A),B),k3_xcmplx_0(A,D))) ) ) ) ) ) ), file(xreal_1,t180_xreal_1), [interesting(0.00)]). fof(l58_borsuk_5,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,A) & ~ r1_xreal_0(D,0) & ~ r1_xreal_0(1,D) & r1_xreal_0(A,k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(1,D),B),k3_xcmplx_0(D,C))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t180_xreal_1]), [file(borsuk_5,l58_borsuk_5),interesting(0.00)]). fof(t179_xreal_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,0) & ~ r1_xreal_0(1,A) & r1_xreal_0(B,C) & ~ r1_xreal_0(D,B) & r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(1,A),C),k3_xcmplx_0(A,D)),B) ) ) ) ) ) ), file(xreal_1,t179_xreal_1), [interesting(0.00)]). fof(l57_borsuk_5,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,A) & ~ r1_xreal_0(D,0) & ~ r1_xreal_0(1,D) & r1_xreal_0(k2_xcmplx_0(k3_xcmplx_0(k6_xcmplx_0(1,D),B),k3_xcmplx_0(D,C)),A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t179_xreal_1]), [file(borsuk_5,l57_borsuk_5),interesting(0.00)]). fof(d2_rat_1,definition,( ! [A] : ( v1_rat_1(A) <=> r2_hidden(A,k3_numbers) ) ), file(rat_1,d2_rat_1), [interesting(0.00)]). fof(d4_xboole_0,definition,( ! [A,B,C] : ( C = k4_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & ~ r2_hidden(D,B) ) ) ) ), file(xboole_0,d4_xboole_0), [interesting(0.00)]). fof(t11_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(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), [interesting(0.00)]). fof(t50_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => r1_xreal_0(0,k6_xcmplx_0(B,A)) ) ) ) ), file(xreal_1,t50_xreal_1), [interesting(0.00)]). fof(t15_topmetr,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k8_metric_1)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k8_metric_1)) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( C = A & D = B ) => k4_metric_1(k8_metric_1,A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) ) ), file(topmetr,t15_topmetr), [interesting(0.00)]). fof(d1_absvalue,definition,( ! [A] : ( v1_xreal_0(A) => ( ( r1_xreal_0(0,A) => k16_complex1(A) = A ) & ( ~ r1_xreal_0(0,A) => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) ), file(absvalue,d1_absvalue), [interesting(0.00)]). fof(t12_metric_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( l1_metric_1(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => ( r2_hidden(D,k9_metric_1(B,C,A)) <=> ( ~ v3_struct_0(B) & ~ r1_xreal_0(A,k2_metric_1(B,C,D)) ) ) ) ) ) ) ), file(metric_1,t12_metric_1), [interesting(0.00)]). fof(t3_xboole_0,theorem,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ ( r2_hidden(C,A) & r2_hidden(C,B) ) ) & ~ ( ? [C] : ( r2_hidden(C,A) & r2_hidden(C,B) ) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t3_xboole_0), [interesting(0.00)]). fof(t7_hausdorf,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r2_hidden(C,k6_pre_topc(k5_pcomps_1(A),B)) <=> ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(D,0) & r1_xboole_0(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ), file(hausdorf,t7_hausdorf), [interesting(0.00)]). fof(t82_topreal6,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( A != B <=> k7_pscomp_1(k2_rcomp_1(A,B)) = k1_rcomp_1(A,B) ) ) ) ), file(topreal6,t82_topreal6), [interesting(0.00)]). fof(t35_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => r1_xreal_0(k1_square_1(A,B),A) ) ) ), file(square_1,t35_square_1), [interesting(0.00)]). fof(t3_topreal3,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => ( ~ r1_xreal_0(k7_xcmplx_0(k2_xcmplx_0(A,B),2),A) & ~ r1_xreal_0(B,k7_xcmplx_0(k2_xcmplx_0(A,B),2)) ) ) ) ) ), file(topreal3,t3_topreal3), [interesting(0.00)]). fof(t38_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( k1_square_1(A,B) = A | k1_square_1(A,B) = B ) ) ) ), file(square_1,t38_square_1), [interesting(0.00)]). fof(t47_rcomp_1,theorem,( ! [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), [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(t46_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => r1_xreal_0(A,k2_square_1(A,B)) ) ) ), file(square_1,t46_square_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(t49_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( k2_square_1(A,B) = A | k2_square_1(A,B) = B ) ) ) ), file(square_1,t49_square_1), [interesting(0.00)]). fof(t105_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ~ ( ~ r1_xreal_0(k2_xcmplx_0(A,k4_xcmplx_0(B)),0) & ~ r1_xreal_0(0,k6_xcmplx_0(B,A)) & ~ r1_xreal_0(0,k2_xcmplx_0(B,k4_xcmplx_0(A))) & ~ r1_xreal_0(k6_xcmplx_0(A,C),k2_xcmplx_0(B,k4_xcmplx_0(C))) & ~ r1_xreal_0(k2_xcmplx_0(A,k4_xcmplx_0(C)),k6_xcmplx_0(B,C)) & ~ r1_xreal_0(k6_xcmplx_0(C,B),k6_xcmplx_0(C,A)) ) => r1_xreal_0(A,B) ) ) ) ) ), file(real_2,t105_real_2), [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(t46_zfmisc_1,theorem,( ! [A,B] : ( r2_hidden(A,B) => k2_xboole_0(k1_tarski(A),B) = B ) ), file(zfmisc_1,t46_zfmisc_1), [interesting(0.00)]). fof(t41_xboole_1,theorem,( ! [A,B,C] : k4_xboole_0(k4_xboole_0(A,B),C) = k4_xboole_0(A,k2_xboole_0(B,C)) ), file(xboole_1,t41_xboole_1), [interesting(0.00)]). fof(t24_limfunc1,theorem,( ! [A] : ( v1_xreal_0(A) => ( k4_xboole_0(k1_numbers,k4_limfunc1(A)) = k2_limfunc1(A) & k4_xboole_0(k1_numbers,k3_limfunc1(A)) = k12_prob_1(A) & k4_xboole_0(k1_numbers,k12_prob_1(A)) = k3_limfunc1(A) & k4_xboole_0(k1_numbers,k2_limfunc1(A)) = k4_limfunc1(A) ) ) ), file(limfunc1,t24_limfunc1), [interesting(0.00)]). fof(t64_zfmisc_1,theorem,( ! [A,B,C] : ( r2_hidden(A,k4_xboole_0(B,k1_tarski(C))) <=> ( r2_hidden(A,B) & A != C ) ) ), file(zfmisc_1,t64_zfmisc_1), [interesting(0.00)]). fof(d3_limfunc1,definition,( ! [A] : ( v1_xreal_0(A) => k4_limfunc1(A) = a_1_2_limfunc1(A) ) ), file(limfunc1,d3_limfunc1), [interesting(0.00)]). fof(t148_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ~ r1_xreal_0(A,k6_xcmplx_0(A,1)) ) ), file(xreal_1,t148_xreal_1), [interesting(0.00)]). fof(d2_limfunc1,definition,( ! [A] : ( v1_xreal_0(A) => k3_limfunc1(A) = a_1_1_limfunc1(A) ) ), file(limfunc1,d2_limfunc1), [interesting(0.00)]). fof(t105_borsuk_5,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))) => ~ ( B != C & k3_subset_1(u1_struct_0(A),B) = k3_subset_1(u1_struct_0(A),C) ) ) ) ) ), file(borsuk_5,t105_borsuk_5), [interesting(0.00)]). fof(t12_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = u1_struct_0(A) ) ), file(pre_topc,t12_pre_topc), [interesting(0.00)]). fof(t27_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = k3_subset_1(u1_struct_0(A),k1_pre_topc(A)) ) ), file(pre_topc,t27_pre_topc), [interesting(0.00)]). fof(d2_borsuk_5,definition,( ! [A,B,C,D,E,F,G] : ( r2_borsuk_5(A,B,C,D,E,F,G) <=> ( A != B & A != C & A != D & A != E & A != F & A != G & B != C & B != D & B != E & B != F & B != G & C != D & C != E & C != F & C != G & D != E & D != F & D != G & E != F & E != G & F != G ) ) ), file(borsuk_5,d2_borsuk_5), [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(t81_topreal6,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ( A = B => ( v1_rcomp_1(A) <=> v6_compts_1(B,k3_topmetr) ) ) ) ) ), file(topreal6,t81_topreal6), [interesting(0.00)]). fof(t28_rcomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_rcomp_1(A) => v3_seq_4(A) ) ) ), file(rcomp_1,t28_rcomp_1), [interesting(0.00)]). fof(d3_seq_4,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v3_seq_4(A) <=> ( v2_seq_4(A) & v1_seq_4(A) ) ) ) ), file(seq_4,d3_seq_4), [interesting(0.00)]). fof(t31_rcomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( ( v2_rcomp_1(A) & v2_seq_4(A) ) => ( A = k1_xboole_0 | r2_hidden(k5_seq_4(A),A) ) ) ) ), file(rcomp_1,t31_rcomp_1), [interesting(0.00)]). fof(t30_rcomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( ( v2_rcomp_1(A) & v1_seq_4(A) ) => ( A = k1_xboole_0 | r2_hidden(k4_seq_4(A),A) ) ) ) ), file(rcomp_1,t30_rcomp_1), [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(t40_zfmisc_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => ( r2_hidden(C,A) | r1_tarski(A,k4_xboole_0(B,k1_tarski(C))) ) ) ), file(zfmisc_1,t40_zfmisc_1), [interesting(0.00)]). fof(t14_rcomp_1,theorem,( ! [A] : ( v1_xreal_0(A) => k1_rcomp_1(A,A) = k1_seq_4(A) ) ), file(rcomp_1,t14_rcomp_1), [interesting(0.00)]). fof(t25_limfunc1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( k4_xboole_0(k1_numbers,k2_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k2_limfunc1(A),k3_limfunc1(B)) & k4_xboole_0(k1_numbers,k1_rcomp_1(A,B)) = k4_subset_1(k1_numbers,k12_prob_1(A),k4_limfunc1(B)) ) ) ) ), file(limfunc1,t25_limfunc1), [interesting(0.00)]). fof(t79_xboole_1,theorem,( ! [A,B] : r1_xboole_0(k4_xboole_0(A,B),B) ), file(xboole_1,t79_xboole_1), [interesting(0.00)]). fof(t7_fcont_3,theorem,( ! [A] : ( v1_xreal_0(A) => v3_rcomp_1(k4_limfunc1(A)) ) ), file(fcont_3,t7_fcont_3), [interesting(0.00)]). fof(t5_fcont_3,theorem,( ! [A] : ( v1_xreal_0(A) => v2_rcomp_1(k3_limfunc1(A)) ) ), file(fcont_3,t5_fcont_3), [interesting(0.00)]). fof(t10_limfunc1,theorem,( ! [A] : ( v1_xreal_0(A) => r1_tarski(k4_limfunc1(A),k3_limfunc1(A)) ) ), file(limfunc1,t10_limfunc1), [interesting(0.00)]). fof(d1_connsp_1,definition,( ! [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))) => ( r1_connsp_1(A,B,C) <=> ( r1_xboole_0(k6_pre_topc(A,B),C) & r1_xboole_0(B,k6_pre_topc(A,C)) ) ) ) ) ) ), file(connsp_1,d1_connsp_1), [interesting(0.00)]). fof(t79_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [C] : ( v1_xreal_0(C) => ( ( A = k12_prob_1(C) & B = k4_limfunc1(C) ) => r1_connsp_1(k3_topmetr,A,B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t77_borsuk_5,t26_borsuk_5,t75_borsuk_5,t27_borsuk_5,d1_connsp_1]), [file(borsuk_5,t79_borsuk_5),interesting(0.00)]). fof(t17_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(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))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v2_connsp_1(B,A) & r1_tarski(B,k4_subset_1(u1_struct_0(A),C,D)) & r1_connsp_1(A,C,D) & ~ r1_tarski(B,C) & ~ r1_tarski(B,D) ) ) ) ) ) ), file(connsp_1,t17_connsp_1), [interesting(0.00)]). fof(d4_seq_4,definition,( ! [A] : ( v2_membered(A) => ( v1_seq_4(A) => ( v1_xboole_0(A) | ! [B] : ( v1_xreal_0(B) => ( B = k2_seq_4(A) <=> ( ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(C,B) ) ) & ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( v1_xreal_0(D) => ~ ( r2_hidden(D,A) & ~ r1_xreal_0(D,k6_xcmplx_0(B,C)) ) ) ) ) ) ) ) ) ) ) ), file(seq_4,d4_seq_4), [interesting(0.00)]). fof(d5_seq_4,definition,( ! [A] : ( v2_membered(A) => ( v2_seq_4(A) => ( v1_xboole_0(A) | ! [B] : ( v1_xreal_0(B) => ( B = k3_seq_4(A) <=> ( ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(B,C) ) ) & ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & ! [D] : ( v1_xreal_0(D) => ~ ( r2_hidden(D,A) & ~ r1_xreal_0(k2_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ) ) ), file(seq_4,d5_seq_4), [interesting(0.00)]). fof(t108_borsuk_5,theorem,( ! [A] : ( ( v6_compts_1(A,k3_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_numbers)) => ! [C] : ( v1_xreal_0(C) => ( ( r2_hidden(C,B) & B = A ) => ( r1_xreal_0(k4_pscomp_1(B),C) & r1_xreal_0(C,k3_pscomp_1(B)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t107_borsuk_5,d4_seq_4,d5_seq_4]), [file(borsuk_5,t108_borsuk_5),interesting(0.00)]). fof(t24_seq_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v3_seq_4(A) => ( v1_xboole_0(A) | r1_xreal_0(k5_seq_4(A),k4_seq_4(A)) ) ) ) ), file(seq_4,t24_seq_4), [interesting(0.00)]). fof(d1_integra1,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_integra1(A) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & ? [C] : ( m1_subset_1(C,k1_numbers) & r1_xreal_0(B,C) & A = k1_rcomp_1(B,C) ) ) ) ) ), file(integra1,d1_integra1), [interesting(0.00)]). fof(t5_integra1,theorem,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => A = k1_rcomp_1(k4_pscomp_1(A),k3_pscomp_1(A)) ) ), file(integra1,t5_integra1), [interesting(0.00)]). fof(t53_borsuk_4,theorem,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => r1_xreal_0(k4_pscomp_1(A),k3_pscomp_1(A)) ) ), file(borsuk_4,t53_borsuk_4), [interesting(0.00)]). fof(t15_rcomp_1,theorem,( ! [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), [interesting(0.00)]). fof(d6_borsuk_5,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) => ( v1_borsuk_5(B,A) <=> ~ r2_hidden(u1_struct_0(A),B) ) ) ) ), file(borsuk_5,d6_borsuk_5), [interesting(0.00)]). fof(t115_borsuk_5,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_borsuk_5(B,A) & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) => ! [C] : ( ( v1_borsuk_5(C,A) & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) => v1_borsuk_5(k4_subset_1(k1_zfmisc_1(u1_struct_0(A)),B,C),A) ) ) ) ), inference(mizar_proof,[status(thm)],[d6_borsuk_5,d2_xboole_0,d6_borsuk_5]), [file(borsuk_5,t115_borsuk_5),interesting(0.00)]). fof(t12_borsuk_5,theorem,( $true ), file(borsuk_5,t12_borsuk_5), [interesting(0.00)]). fof(t1_borsuk_5,theorem,( $true ), file(borsuk_5,t1_borsuk_5), [interesting(0.00)]). fof(t21_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( A != B => k7_pscomp_1(k1_rcomp_2(A,B)) = k1_rcomp_1(A,B) ) ) ) ), file(borsuk_4,t21_borsuk_4), [interesting(0.00)]). fof(t4_rcomp_2,theorem,( ! [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), [interesting(0.00)]). fof(t3_rcomp_2,theorem,( ! [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), [interesting(0.00)]). fof(t21_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ! [E] : ( v1_xreal_0(E) => ! [F] : ( v1_xreal_0(F) => ( ( r1_xreal_0(D,E) & A = k1_rcomp_2(C,D) & B = k2_rcomp_2(E,F) ) => ( r1_xreal_0(D,C) | r1_xreal_0(F,E) | r1_connsp_1(k3_topmetr,A,B) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t21_borsuk_4,t80_topreal6,t19_borsuk_5,t20_borsuk_4,t80_topreal6,t20_borsuk_5,d1_connsp_1]), [file(borsuk_5,t21_borsuk_5),interesting(0.00)]). fof(t22_borsuk_5,theorem,( $true ), file(borsuk_5,t22_borsuk_5), [interesting(0.00)]). fof(t23_borsuk_5,theorem,( $true ), file(borsuk_5,t23_borsuk_5), [interesting(0.00)]). fof(t24_borsuk_5,theorem,( $true ), file(borsuk_5,t24_borsuk_5), [interesting(0.00)]). fof(t25_borsuk_5,theorem,( $true ), file(borsuk_5,t25_borsuk_5), [interesting(0.00)]). fof(t30_borsuk_5,theorem,( $true ), file(borsuk_5,t30_borsuk_5), [interesting(0.00)]). fof(t31_borsuk_5,theorem,( $true ), file(borsuk_5,t31_borsuk_5), [interesting(0.00)]). fof(t32_borsuk_5,theorem,( $true ), file(borsuk_5,t32_borsuk_5), [interesting(0.00)]). fof(t33_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k8_metric_1)) => ( r2_hidden(B,k6_pre_topc(k3_topmetr,A)) <=> ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & r1_xboole_0(k9_metric_1(k8_metric_1,B,C),A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t7_hausdorf,d7_topmetr]), [file(borsuk_5,t33_borsuk_5),interesting(0.00)]). fof(d4_enumset1,definition,( ! [A,B,C,D,E,F,G] : ( G = k4_enumset1(A,B,C,D,E,F) <=> ! [H] : ( r2_hidden(H,G) <=> ~ ( H != A & H != B & H != C & H != D & H != E & H != F ) ) ) ), file(enumset1,d4_enumset1), [interesting(0.00)]). fof(d1_enumset1,definition,( ! [A,B,C,D] : ( D = k1_enumset1(A,B,C) <=> ! [E] : ( r2_hidden(E,D) <=> ~ ( E != A & E != B & E != C ) ) ) ), file(enumset1,d1_enumset1), [interesting(0.00)]). fof(t39_borsuk_5,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v1_rat_1(A) ) => ~ v1_rat_1(k4_xcmplx_0(A)) ) ), file(borsuk_5,t39_borsuk_5), [interesting(0.00)]). fof(t88_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ), file(xcmplx_1,t88_xcmplx_1), [interesting(0.00)]). fof(t50_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ~ ( A != 0 & B != 0 & k7_xcmplx_0(A,B) = 0 ) ) ) ), file(xcmplx_1,t50_xcmplx_1), [interesting(0.00)]). fof(d6_int_1,definition,( ! [A] : ( v1_xreal_0(A) => k3_int_1(A) = k6_xcmplx_0(A,k1_int_1(A)) ) ), file(int_1,d6_int_1), [interesting(0.00)]). fof(t46_borsuk_5,theorem,( $true ), file(borsuk_5,t46_borsuk_5), [interesting(0.00)]). fof(t48_borsuk_5,theorem,( $true ), file(borsuk_5,t48_borsuk_5), [interesting(0.00)]). fof(t49_borsuk_5,theorem,( $true ), file(borsuk_5,t49_borsuk_5), [interesting(0.00)]). fof(d1_borsuk_5,definition,( ! [A,B,C,D,E,F] : ( r1_borsuk_5(A,B,C,D,E,F) <=> ( A != B & A != C & A != D & A != E & A != F & B != C & B != D & B != E & B != F & C != D & C != E & C != F & D != E & D != F & E != F ) ) ), file(borsuk_5,d1_borsuk_5), [interesting(0.00)]). fof(d1_necklace,definition,( ! [A,B,C,D,E] : ( r1_necklace(A,B,C,D,E) <=> ( A != B & A != C & A != D & A != E & B != C & B != D & B != E & C != D & C != E & D != E ) ) ), file(necklace,d1_necklace), [interesting(0.00)]). fof(t1_necklace,theorem,( ! [A,B,C,D,E] : ( r1_necklace(A,B,C,D,E) => k4_card_1(k3_enumset1(A,B,C,D,E)) = 5 ) ), file(necklace,t1_necklace), [interesting(0.00)]). fof(d3_enumset1,definition,( ! [A,B,C,D,E,F] : ( F = k3_enumset1(A,B,C,D,E) <=> ! [G] : ( r2_hidden(G,F) <=> ~ ( G != A & G != B & G != C & G != D & G != E ) ) ) ), file(enumset1,d3_enumset1), [interesting(0.00)]). fof(t55_enumset1,theorem,( ! [A,B,C,D,E,F] : k4_enumset1(A,B,C,D,E,F) = k2_xboole_0(k3_enumset1(A,B,C,D,E),k1_tarski(F)) ), file(enumset1,t55_enumset1), [interesting(0.00)]). fof(t54_card_2,theorem,( ! [A,B] : ( v1_finset_1(B) => ( ~ r2_hidden(A,B) => k4_card_1(k2_xboole_0(B,k1_tarski(A))) = k1_nat_1(k4_card_1(B),1) ) ) ), file(card_2,t54_card_2), [interesting(0.00)]). fof(t61_enumset1,theorem,( ! [A,B,C,D,E,F,G] : k5_enumset1(A,B,C,D,E,F,G) = k2_xboole_0(k4_enumset1(A,B,C,D,E,F),k1_tarski(G)) ), file(enumset1,t61_enumset1), [interesting(0.00)]). fof(t2_treal_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(B,C) ) => k4_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_1(A,D) ) ) ) ) ) ), file(treal_1,t2_treal_1), [interesting(0.00)]). fof(t69_borsuk_5,theorem,( $true ), file(borsuk_5,t69_borsuk_5), [interesting(0.00)]). fof(t70_borsuk_5,theorem,( $true ), file(borsuk_5,t70_borsuk_5), [interesting(0.00)]). fof(d5_incproj,definition,( ! [A,B,C] : ( r1_incproj(A,B,C) <=> ( A != B & B != C & C != A ) ) ), file(incproj,d5_incproj), [interesting(0.00)]). fof(t12_limfunc1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => r1_tarski(k1_rcomp_1(A,B),k3_limfunc1(A)) ) ) ), file(limfunc1,t12_limfunc1), [interesting(0.00)]). fof(t9_limfunc1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => r1_tarski(k3_limfunc1(B),k3_limfunc1(A)) ) ) ) ), file(limfunc1,t9_limfunc1), [interesting(0.00)]). fof(t4_xboole_1,theorem,( ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ), file(xboole_1,t4_xboole_1), [interesting(0.00)]). fof(t17_xboole_1,theorem,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_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(t12_numbers,theorem,( r1_tarski(k3_numbers,k1_numbers) ), file(numbers,t12_numbers), [interesting(0.00)]). fof(t12_rcomp_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k2_rcomp_1(B,A) = k1_xboole_0 ) ) ) ), file(rcomp_1,t12_rcomp_1), [interesting(0.00)]). fof(t12_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k2_xboole_0(A,B) = B ) ), file(xboole_1,t12_xboole_1), [interesting(0.00)]). fof(t64_xboole_1,theorem,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) & r1_xboole_0(B,D) ) => r1_xboole_0(A,C) ) ), file(xboole_1,t64_xboole_1), [interesting(0.00)]). fof(t87_xboole_1,theorem,( ! [A,B,C] : ( r1_xboole_0(A,B) => k2_xboole_0(k4_xboole_0(C,A),B) = k4_xboole_0(k2_xboole_0(C,B),A) ) ), file(xboole_1,t87_xboole_1), [interesting(0.00)]). fof(t57_zfmisc_1,theorem,( ! [A,B,C] : ~ ( ~ r2_hidden(A,B) & ~ r2_hidden(C,B) & ~ r1_xboole_0(k2_tarski(A,C),B) ) ), file(zfmisc_1,t57_zfmisc_1), [interesting(0.00)]). fof(t11_rcomp_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k1_rcomp_1(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k2_tarski(A,B)) ) ) ) ), file(rcomp_1,t11_rcomp_1), [interesting(0.00)]). fof(t14_zfmisc_1,theorem,( ! [A,B] : k2_xboole_0(k1_tarski(A),k2_tarski(A,B)) = k2_tarski(A,B) ), file(zfmisc_1,t14_zfmisc_1), [interesting(0.00)]). fof(t88_xboole_1,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => k4_xboole_0(k2_xboole_0(A,B),B) = A ) ), file(xboole_1,t88_xboole_1), [interesting(0.00)]). fof(t41_enumset1,theorem,( ! [A,B] : k2_tarski(A,B) = k2_xboole_0(k1_tarski(A),k1_tarski(B)) ), file(enumset1,t41_enumset1), [interesting(0.00)]). fof(t88_borsuk_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => k6_subset_1(k1_numbers,k3_limfunc1(A),k4_subset_1(k1_numbers,k2_rcomp_1(A,B),k4_limfunc1(B))) = k4_subset_1(k1_numbers,k1_seq_4(A),k1_seq_4(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t86_borsuk_5,t87_borsuk_5,t57_zfmisc_1,t28_borsuk_5,t11_rcomp_1,t67_borsuk_5,t4_xboole_1,t4_xboole_1,t14_zfmisc_1,t4_xboole_1,t88_xboole_1,t41_enumset1]), [file(borsuk_5,t88_borsuk_5),interesting(0.00)]). fof(t54_zfmisc_1,theorem,( ! [A,B] : ~ ( r1_xboole_0(k1_tarski(A),B) & r2_hidden(A,B) ) ), file(zfmisc_1,t54_zfmisc_1), [interesting(0.00)]). fof(t90_borsuk_5,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => ! [B] : ( v1_xreal_0(B) => ( A = k1_seq_4(B) => k3_subset_1(u1_struct_0(k3_topmetr),A) = k4_subset_1(k1_numbers,k12_prob_1(B),k4_limfunc1(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_borsuk_5,t17_pre_topc,t18_borsuk_5]), [file(borsuk_5,t90_borsuk_5),interesting(0.00)]). fof(t63_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_xboole_0(B,C) ) => r1_xboole_0(A,C) ) ), file(xboole_1,t63_xboole_1), [interesting(0.00)]). fof(t113_xboole_1,theorem,( ! [A,B,C,D] : k2_xboole_0(k2_xboole_0(k2_xboole_0(A,B),C),D) = k2_xboole_0(A,k2_xboole_0(k2_xboole_0(B,C),D)) ), file(xboole_1,t113_xboole_1), [interesting(0.00)]). fof(t78_xboole_1,theorem,( ! [A,B,C] : ( r1_xboole_0(A,B) => k3_xboole_0(A,k2_xboole_0(B,C)) = k3_xboole_0(A,C) ) ), file(xboole_1,t78_xboole_1), [interesting(0.00)]). fof(l115_borsuk_5,theorem,( k5_subset_1(k1_numbers,k12_prob_1(1),k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_limfunc1(2),k3_borsuk_5(2,4)),k1_seq_4(4)),k1_seq_4(5))) = k12_prob_1(1) ), inference(mizar_proof,[status(thm)],[t27_borsuk_5,l114_borsuk_5,t63_xboole_1,d3_tarski,d1_xreal_0,t17_borsuk_5,t2_xreal_1,t16_borsuk_5,t113_xboole_1,t78_xboole_1,t28_xboole_1]), [file(borsuk_5,l115_borsuk_5),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(l117_borsuk_5,theorem,( k5_subset_1(k1_numbers,k4_limfunc1(1),k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_limfunc1(2),k3_borsuk_5(2,4)),k1_seq_4(4)),k1_seq_4(5))) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_rcomp_2(1,2),k3_borsuk_5(2,4)),k1_seq_4(4)),k1_seq_4(5)) ), inference(mizar_proof,[status(thm)],[t113_xboole_1,t23_xboole_1,l112_borsuk_5,t28_xboole_1,t91_borsuk_5,t113_xboole_1]), [file(borsuk_5,l117_borsuk_5),interesting(0.00)]). fof(t92_borsuk_5,theorem,( k5_subset_1(k1_numbers,k4_subset_1(k1_numbers,k12_prob_1(1),k4_limfunc1(1)),k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k2_limfunc1(2),k3_borsuk_5(2,4)),k1_seq_4(4)),k1_seq_4(5))) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k12_prob_1(1),k2_rcomp_2(1,2)),k3_borsuk_5(2,4)),k1_seq_4(4)),k1_seq_4(5)) ), inference(mizar_proof,[status(thm)],[l115_borsuk_5,t23_xboole_1,l117_borsuk_5,t4_xboole_1,t113_xboole_1,t4_xboole_1]), [file(borsuk_5,t92_borsuk_5),interesting(0.00)]).