fof(t77_borsuk_4,theorem,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( r2_hidden(B,A) => r1_t_0topsp(k3_pre_topc(k15_euclid(2),k6_subset_1(u1_struct_0(k15_euclid(2)),A,k1_struct_0(k15_euclid(2),B))),k1_borsuk_4) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_borsuk_4,t5_topreal2,d1_tarski,d4_xboole_0,t42_xboole_1,t14_jordan5b,t76_borsuk_4]), [file(borsuk_4,t77_borsuk_4),interesting(1.00)]). fof(t72_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ( A = k2_rcomp_2(0,B) => v4_pre_topc(A,k1_borsuk_4) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,t2_jordan5a,t4_rcomp_2,t60_borsuk_4,d5_real_1,t12_pre_topc,d1_borsuk_4,t41_borsuk_4,t71_borsuk_4,d6_pre_topc,t10_rcomp_2]), [file(borsuk_4,t72_borsuk_4),interesting(0.94)]). fof(t73_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ( A = k1_rcomp_2(B,1) => v4_pre_topc(A,k1_borsuk_4) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,t7_rcomp_2,t2_jordan5a,d5_real_1,t12_pre_topc,d1_borsuk_4,t3_rcomp_2,d1_borsuk_4,t47_rcomp_1,t2_jordan5a,t42_borsuk_4,t71_borsuk_4,d6_pre_topc]), [file(borsuk_4,t73_borsuk_4),interesting(0.94)]). fof(t55_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k5_topmetr) & v6_compts_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) ) ), inference(mizar_proof,[status(thm)],[t83_borsuk_1,t1_xboole_1,t16_compts_1,t52_borsuk_4,t50_borsuk_4,t31_rcomp_1,t30_rcomp_1,t47_borsuk_4,t54_borsuk_4,d3_seq_4,t24_seq_4,d1_integra1]), [file(borsuk_4,t55_borsuk_4),interesting(0.92)]). fof(t3_borsuk_4,theorem,( ! [A] : ( ~ v1_realset1(A) => ! [B] : ~ ! [C] : ( m1_subset_1(C,A) => C = B ) ) ), inference(mizar_proof,[status(thm)],[t4_realset1,d1_xboole_0,d4_xboole_0,t64_zfmisc_1]), [file(borsuk_4,t3_borsuk_4),interesting(0.92)]). fof(t84_borsuk_4,theorem,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( ( v4_pre_topc(B,k15_euclid(2)) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( r1_tarski(B,A) & B != A & ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ~ ( r1_topreal1(k15_euclid(2),C,D,E) & r1_tarski(B,E) & r1_tarski(E,A) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t73_jordan2c,t16_jordan2c,t88_topreal6,t15_realset1,d8_xboole_0,t1_borsuk_4,t77_borsuk_4,d1_t_0topsp,t1_jordan1,t1_jordan1,t12_pre_topc,t11_compts_1,d5_tops_2,d5_tops_2,t24_compts_1,t39_pre_topc,t12_pre_topc,d1_borsuk_4,t1_jordan1,t43_funct_2,t1_jordan1,t43_funct_2,t11_compts_1,d1_funct_2,d1_funct_2,t83_borsuk_4,t49_borsuk_4,d5_tops_2,t14_rcomp_1,d1_tarski,d1_tarski,d8_funct_1,d5_real_1,d3_pre_topc,d1_borsuk_4,t174_relat_1,t167_relat_1,d1_funct_2,t35_borsuk_1,t1_xboole_1,d1_funct_2,t1_jordan1,t36_xboole_1,t1_xboole_1,t147_funct_1,t80_borsuk_4,d1_funct_2,t3_jct_misc,t4_topreal2,d2_tarski,t14_realset1,t41_enumset1,t19_compts_1,t38_zfmisc_1,t37_xboole_1,t2_xboole_1,t60_jordan6,t2_xboole_1,t2_borsuk_4,d1_tarski,t3_borsuk_4,t19_compts_1,t7_xboole_1,d1_tarski,d2_xboole_0,t14_realset1,t37_zfmisc_1,t8_xboole_1,t37_xboole_1,t2_xboole_1,t41_enumset1,t60_jordan6,t1_xboole_1]), [file(borsuk_4,t84_borsuk_4),interesting(0.88)]). fof(t78_borsuk_4,theorem,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ( ( r2_hidden(B,A) & r2_hidden(C,A) ) => r1_t_0topsp(k3_pre_topc(k15_euclid(2),k6_subset_1(u1_struct_0(k15_euclid(2)),A,k1_struct_0(k15_euclid(2),B))),k3_pre_topc(k15_euclid(2),k6_subset_1(u1_struct_0(k15_euclid(2)),A,k1_struct_0(k15_euclid(2),C)))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_borsuk_4,t64_zfmisc_1,t64_zfmisc_1,t64_zfmisc_1,t77_borsuk_4,t77_borsuk_4,t3_borsuk_3]), [file(borsuk_4,t78_borsuk_4),interesting(0.87)]). fof(t47_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_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)],[t2_jordan5a,d3_tarski,d1_rcomp_1,t2_xreal_1,t2_jordan5a,t46_borsuk_4]), [file(borsuk_4,t47_borsuk_4),interesting(0.87)]). fof(t40_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,B) & k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k1_rcomp_2(B,C)) != k2_rcomp_1(A,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t38_borsuk_4,t7_xboole_1,t39_borsuk_4,t8_xboole_1,d10_xboole_0,d3_tarski,d1_xreal_0,t47_rcomp_1,t4_rcomp_2,d2_xboole_0,t3_rcomp_2,d2_xboole_0]), [file(borsuk_4,t40_borsuk_4),interesting(0.86)]). 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) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_rcomp_2,t32_pscomp_1,t17_rcomp_2,t37_pscomp_1,t82_topreal6,d10_xboole_0]), [file(borsuk_4,t21_borsuk_4),interesting(0.86)]). 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) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_rcomp_2,t32_pscomp_1,t17_rcomp_2,t37_pscomp_1,t82_topreal6,d10_xboole_0]), [file(borsuk_4,t20_borsuk_4),interesting(0.86)]). fof(t15_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(C,D)) => ( r1_xreal_0(B,A) | ( r1_xreal_0(C,A) & r1_xreal_0(B,D) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_rcomp_1,t69_xboole_1,t7_xreal_1,t11_borsuk_4,t2_xreal_1,t47_rcomp_1,t48_rcomp_1,t2_rcomp_2,t7_xreal_1,t46_square_1,t2_xreal_1,t2_xreal_1,t47_rcomp_1,t48_rcomp_1]), [file(borsuk_4,t15_borsuk_4),interesting(0.85)]). fof(t41_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,B) & k6_subset_1(k1_numbers,k2_rcomp_1(A,C),k2_rcomp_2(A,B)) != k2_rcomp_1(B,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_rcomp_2,t12_borsuk_4,t63_xboole_1,t38_borsuk_4,t40_xboole_1,t83_xboole_1]), [file(borsuk_4,t41_borsuk_4),interesting(0.85)]). fof(t42_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,B) & k6_subset_1(k1_numbers,k2_rcomp_1(A,C),k1_rcomp_2(B,C)) != k2_rcomp_1(A,B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t38_borsuk_4,t6_rcomp_2,t4_xboole_1,t5_rcomp_2,t17_rcomp_2,t13_borsuk_4,t63_xboole_1,t88_xboole_1]), [file(borsuk_4,t42_borsuk_4),interesting(0.85)]). fof(t59_borsuk_4,theorem, ( v1_tsep_1(k1_borsuk_4,k5_topmetr) & m1_pre_topc(k1_borsuk_4,k5_topmetr) ), inference(mizar_proof,[status(thm)],[t58_borsuk_4,t41_jordan6,d1_tsep_1]), [file(borsuk_4,t59_borsuk_4),interesting(0.84)]). fof(t71_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ! [C] : ( m1_subset_1(C,u1_struct_0(k5_topmetr)) => ( A = k2_rcomp_1(B,C) => v3_pre_topc(A,k1_borsuk_4) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t44_borsuk_4,t70_borsuk_4,t59_borsuk_4,t17_tsep_1]), [file(borsuk_4,t71_borsuk_4),interesting(0.84)]). fof(t39_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,B) & ~ r1_tarski(k1_rcomp_2(B,C),k2_rcomp_1(A,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d1_xreal_0,t3_rcomp_2,t2_xreal_1,t47_rcomp_1]), [file(borsuk_4,t39_borsuk_4),interesting(0.84)]). fof(t4_borsuk_4,theorem,( ! [A] : ( ~ v1_realset1(A) => ! [B] : ( ( ~ v1_realset1(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ? [D] : ( m1_subset_1(D,A) & r2_hidden(D,B) & D != C ) ) ) ), inference(mizar_proof,[status(thm)],[t4_realset1,d1_xboole_0,d4_xboole_0,t64_zfmisc_1]), [file(borsuk_4,t4_borsuk_4),interesting(0.83)]). fof(t38_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,B) & k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_1(B,C)) != k2_rcomp_1(A,C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_xreal_1,l27_borsuk_4,t4_rcomp_2,t56_zfmisc_1,l27_borsuk_4,t70_xboole_1,t12_rcomp_2,t6_rcomp_2,t6_rcomp_2,t4_xboole_1,t88_xboole_1,t88_xboole_1]), [file(borsuk_4,t38_borsuk_4),interesting(0.82)]). fof(t70_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ! [C] : ( m1_subset_1(C,u1_struct_0(k5_topmetr)) => ( A = k2_rcomp_1(B,C) => v3_pre_topc(A,k5_topmetr) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,t2_jordan5a,t83_borsuk_1,t16_rcomp_1,t2_jordan5a,t83_borsuk_1,t16_rcomp_1,t48_borsuk_4,t48_borsuk_4,t36_tops_1,t37_borsuk_4,t83_borsuk_1,t12_pre_topc,t17_pre_topc,t29_tops_1,t12_rcomp_1,t5_pre_topc,d5_pre_topc]), [file(borsuk_4,t70_borsuk_4),interesting(0.82)]). fof(t56_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k5_topmetr) & v6_compts_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ? [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) & ? [C] : ( m1_subset_1(C,u1_struct_0(k5_topmetr)) & r1_xreal_0(B,C) & A = k1_rcomp_1(B,C) ) ) ) ), inference(mizar_proof,[status(thm)],[t55_borsuk_4,t5_integra1,t53_borsuk_4,t15_rcomp_1,t53_borsuk_4,t5_integra1]), [file(borsuk_4,t56_borsuk_4),interesting(0.82)]). fof(t11_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xboole_0(k2_rcomp_1(A,B),k1_rcomp_1(C,D)) & r1_xreal_0(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,d2_rcomp_1,d1_rcomp_1,t2_xreal_1]), [file(borsuk_4,t11_borsuk_4),interesting(0.81)]). fof(t10_borsuk_4,theorem,( ! [A,B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( r2_hidden(A,k1_rcomp_1(B,C)) & ~ r2_hidden(A,k2_rcomp_1(B,C)) & A != B & A != C ) ) ) ), inference(mizar_proof,[status(thm)],[t13_rcomp_1,t11_rcomp_1,d2_xboole_0,d2_tarski]), [file(borsuk_4,t10_borsuk_4),interesting(0.81)]). fof(t26_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xboole_0(k1_rcomp_2(A,B),k1_seq_4(B)) & r1_xboole_0(k2_rcomp_2(A,B),k1_seq_4(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_rcomp_2,t56_zfmisc_1,t4_rcomp_2,t56_zfmisc_1]), [file(borsuk_4,t26_borsuk_4),interesting(0.81)]). fof(l53_borsuk_4,theorem, ( v1_borsuk_1(k5_topmetr,k3_topmetr) & m1_pre_topc(k5_topmetr,k3_topmetr) ), inference(mizar_proof,[status(thm)],[t27_topmetr,t5_treal_1]), [file(borsuk_4,l53_borsuk_4),interesting(0.81)]). fof(l27_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xboole_0(k2_rcomp_1(A,B),k1_seq_4(B)) & r1_xboole_0(k2_rcomp_1(A,B),k1_seq_4(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[t47_rcomp_1,t56_zfmisc_1,t47_rcomp_1,t56_zfmisc_1]), [file(borsuk_4,l27_borsuk_4),interesting(0.80)]). fof(t29_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(C,B) & k5_subset_1(k1_numbers,k2_rcomp_2(A,B),k1_rcomp_2(B,C)) != k1_seq_4(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_rcomp_2,t37_zfmisc_1,t17_rcomp_2,t13_borsuk_4,t63_xboole_1,t6_rcomp_2,t78_xboole_1,t28_xboole_1]), [file(borsuk_4,t29_borsuk_4),interesting(0.79)]). fof(l85_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( k1_rcomp_1(A,B) = k1_rcomp_1(C,D) => ( ( ~ r1_xreal_0(A,B) & ~ r1_xreal_0(C,D) ) | ( A = C & B = D ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t48_rcomp_1,t13_rcomp_1,t20_jordan5a]), [file(borsuk_4,l85_borsuk_4),interesting(0.79)]). fof(t44_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) ) ), inference(mizar_proof,[status(thm)],[t15_rcomp_1,t83_borsuk_1,t16_rcomp_1,t1_xboole_1]), [file(borsuk_4,t44_borsuk_4),interesting(0.78)]). fof(t52_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ( A = B => ( v2_rcomp_1(A) <=> v4_pre_topc(B,k5_topmetr) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t35_borsuk_1,t1_xboole_1,t35_borsuk_1,l53_borsuk_4,d14_borsuk_1,t8_tsep_1,t79_topreal6,t12_pre_topc,t28_xboole_1,t43_pre_topc,t79_topreal6]), [file(borsuk_4,t52_borsuk_4),interesting(0.78)]). fof(t19_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k1_rcomp_2(B,C) => ( r1_xreal_0(C,B) | r1_tarski(k1_rcomp_1(B,C),u1_struct_0(k5_topmetr)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_rcomp_2,t83_borsuk_1,t1_xboole_1,t15_borsuk_4,t15_borsuk_4,t83_borsuk_1,t1_treal_1]), [file(borsuk_4,t19_borsuk_4),interesting(0.78)]). fof(t18_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_rcomp_2(B,C) => ( r1_xreal_0(C,B) | r1_tarski(k1_rcomp_1(B,C),u1_struct_0(k5_topmetr)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t17_rcomp_2,t83_borsuk_1,t1_xboole_1,t15_borsuk_4,t15_borsuk_4,t83_borsuk_1,t1_treal_1]), [file(borsuk_4,t18_borsuk_4),interesting(0.78)]). 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)) ) ), inference(mizar_proof,[status(thm)],[t1_integra2,t2_xreal_1]), [file(borsuk_4,t53_borsuk_4),interesting(0.78)]). fof(t17_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_rcomp_1(B,C) => ( r1_xreal_0(C,B) | r1_tarski(k1_rcomp_1(B,C),u1_struct_0(k5_topmetr)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_borsuk_4,t83_borsuk_1,t15_borsuk_4,t83_borsuk_1,t83_borsuk_1,t1_treal_1]), [file(borsuk_4,t17_borsuk_4),interesting(0.77)]). fof(t27_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k6_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_seq_4(A)) = k2_rcomp_2(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[t26_borsuk_4,t20_rcomp_2,t14_rcomp_1,t40_xboole_1,t83_xboole_1]), [file(borsuk_4,t27_borsuk_4),interesting(0.77)]). fof(t28_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k6_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_seq_4(B)) = k1_rcomp_2(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[t26_borsuk_4,t21_rcomp_2,t14_rcomp_1,t40_xboole_1,t83_xboole_1]), [file(borsuk_4,t28_borsuk_4),interesting(0.77)]). fof(t12_borsuk_4,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_xboole_0(k1_rcomp_1(A,B),k2_rcomp_1(C,D)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,d1_rcomp_1,d2_rcomp_1,t2_xreal_1]), [file(borsuk_4,t12_borsuk_4),interesting(0.76)]). fof(t13_borsuk_4,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_xboole_0(k2_rcomp_1(A,B),k1_rcomp_1(C,D)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,d2_rcomp_1,d1_rcomp_1,t2_xreal_1]), [file(borsuk_4,t13_borsuk_4),interesting(0.76)]). fof(t2_borsuk_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ( v1_realset1(B) <=> ? [C] : ( m1_subset_1(C,A) & B = k1_tarski(C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_tex_2]), [file(borsuk_4,t2_borsuk_4),interesting(0.75)]). fof(t16_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(C,D)) => ( r1_xreal_0(B,A) | r1_tarski(k1_rcomp_1(A,B),k1_rcomp_1(C,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_rcomp_1,t69_xboole_1,d3_tarski,t7_xreal_1,t11_borsuk_4,t2_xreal_1,t47_rcomp_1,t48_rcomp_1,t12_borsuk_4,t48_rcomp_1,t7_xreal_1,t11_borsuk_4,t2_xreal_1,t15_borsuk_4,t48_rcomp_1,t10_borsuk_4]), [file(borsuk_4,t16_borsuk_4),interesting(0.74)]). fof(t58_borsuk_4,theorem,( u1_struct_0(k1_borsuk_4) = k4_xboole_0(u1_struct_0(k5_topmetr),k2_tarski(0,1)) ), inference(mizar_proof,[status(thm)],[d1_borsuk_4,t46_rcomp_1,t11_rcomp_1,t83_borsuk_1,t88_xboole_1]), [file(borsuk_4,t58_borsuk_4),interesting(0.74)]). fof(t57_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ( A = u1_struct_0(k1_borsuk_4) => k1_borsuk_4 = k3_pre_topc(k5_topmetr,A) ) ) ), inference(mizar_proof,[status(thm)],[t1_jordan1,t5_tsep_1]), [file(borsuk_4,t57_borsuk_4),interesting(0.74)]). fof(t83_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ~ ( r1_tarski(A,k2_rcomp_1(0,1)) & ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ! [C] : ( m1_subset_1(C,u1_struct_0(k5_topmetr)) => ~ ( r1_xreal_0(B,C) & r1_tarski(A,k1_rcomp_1(B,C)) & r1_tarski(k1_rcomp_1(B,C),k2_rcomp_1(0,1)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t82_borsuk_4,d1_integra1,t15_rcomp_1,t15_rcomp_1,t83_borsuk_1]), [file(borsuk_4,t83_borsuk_4),interesting(0.74)]). fof(t25_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & k1_rcomp_1(A,B) = k2_rcomp_2(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_rcomp_2,t48_rcomp_1]), [file(borsuk_4,t25_borsuk_4),interesting(0.73)]). fof(t50_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_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)],[t2_jordan5a,d1_seq_4,t2_jordan5a,d2_seq_4]), [file(borsuk_4,t50_borsuk_4),interesting(0.73)]). fof(t1_borsuk_4,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(A)) ) => ~ ( r2_xboole_0(B,C) & ! [D] : ( m1_subset_1(D,A) => ~ ( r2_hidden(D,C) & r1_tarski(B,k4_xboole_0(C,k1_tarski(D))) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t105_xboole_1,d1_xboole_0,d4_xboole_0,d8_xboole_0,t40_zfmisc_1]), [file(borsuk_4,t1_borsuk_4),interesting(0.72)]). fof(t81_borsuk_4,theorem,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( ( ~ v1_xboole_0(B) & v2_connsp_1(B,k15_euclid(2)) & v6_compts_1(B,k15_euclid(2)) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( r1_tarski(B,A) & B != A & ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ~ r1_topreal1(k15_euclid(2),C,D,B) ) ) & ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => B != k1_struct_0(k15_euclid(2),C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_borsuk_4,t15_realset1,d8_xboole_0,t1_borsuk_4,t77_borsuk_4,d1_t_0topsp,t1_jordan1,t1_jordan1,t12_pre_topc,t11_compts_1,t24_connsp_1,d5_tops_2,d5_tops_2,t24_compts_1,t5_topreal5,t39_pre_topc,t12_pre_topc,t1_jordan1,t43_funct_2,t1_jordan1,t43_funct_2,t11_compts_1,t24_connsp_1,d1_funct_2,d1_funct_2,t56_borsuk_4,d5_tops_2,t14_rcomp_1,d1_tarski,d1_tarski,d8_funct_1,d5_real_1,t80_borsuk_4]), [file(borsuk_4,t81_borsuk_4),interesting(0.72)]). fof(t48_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ( C = k1_rcomp_1(A,B) => v4_pre_topc(C,k5_topmetr) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t32_borsuk_4,t6_treal_1,t35_borsuk_1,d14_borsuk_1,t25_topmetr,t27_topmetr,t13_rcomp_1]), [file(borsuk_4,t48_borsuk_4),interesting(0.70)]). fof(t6_borsuk_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v2_funct_1(A) & v2_funct_1(B) & k3_xboole_0(k1_relat_1(A),k1_relat_1(B)) = k1_tarski(C) & k3_xboole_0(k2_relat_1(A),k2_relat_1(B)) = k1_tarski(k1_funct_1(A,C)) & k1_funct_1(A,C) = k1_funct_1(B,C) ) => k2_funct_1(k1_funct_4(A,B)) = k1_funct_4(k2_funct_1(A),k2_funct_1(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_tarski,d6_partfun1,t55_funct_1,t17_xboole_1,t37_zfmisc_1,t17_xboole_1,t37_zfmisc_1,d1_tarski,t54_funct_1,d1_tarski,t54_funct_1,d6_partfun1,t5_borsuk_4,t5_circcmb2,d1_funct_4,t55_funct_1,t55_funct_1,t5_circcmb2,d1_funct_4,t7_xboole_1,t7_xboole_1,t55_funct_1,t16_funct_4,t54_funct_1,t16_funct_4,t55_funct_1,t14_funct_4,t54_funct_1,t14_funct_4,t13_funct_4,t16_funct_4,t54_funct_1,t12_funct_1,t5_circcmb2,t7_xboole_1,t55_funct_1,t16_funct_4,t14_funct_4,t54_funct_1,t12_funct_1,t7_xboole_1,t55_funct_1,t14_funct_4,t13_funct_4,t54_funct_1]), [file(borsuk_4,t6_borsuk_4),interesting(0.70)]). fof(t30_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xboole_0(k1_rcomp_2(A,B),k1_rcomp_1(B,C)) & r1_xboole_0(k1_rcomp_1(A,B),k2_rcomp_2(B,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_xboole_0,t3_rcomp_2,t48_rcomp_1,t3_xboole_0,t4_rcomp_2,t48_rcomp_1]), [file(borsuk_4,t30_borsuk_4),interesting(0.67)]). fof(t51_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_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)],[t50_borsuk_4,d4_seq_4,d5_seq_4]), [file(borsuk_4,t51_borsuk_4),interesting(0.67)]). fof(t14_borsuk_4,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_tarski(k1_rcomp_1(A,B),k1_rcomp_1(C,D)) ) => ( r1_xreal_0(C,A) & r1_xreal_0(B,D) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_rcomp_1,t15_rcomp_1,t48_rcomp_1]), [file(borsuk_4,t14_borsuk_4),interesting(0.66)]). fof(t32_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(B,C) & A = k1_rcomp_1(B,C) ) => ( r1_xreal_0(0,B) & r1_xreal_0(C,1) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_rcomp_1,t2_jordan5a]), [file(borsuk_4,t32_borsuk_4),interesting(0.66)]). fof(t45_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_integra1(k1_seq_4(A)) & m1_subset_1(k1_seq_4(A),k1_zfmisc_1(k1_numbers)) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,t14_rcomp_1,d1_integra1]), [file(borsuk_4,t45_borsuk_4),interesting(0.65)]). fof(t60_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ( r2_hidden(A,u1_struct_0(k1_borsuk_4)) <=> ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(1,A) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_borsuk_4,t47_rcomp_1,t47_rcomp_1,d1_borsuk_4]), [file(borsuk_4,t60_borsuk_4),interesting(0.64)]). fof(t24_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k1_rcomp_2(B,C) => ( r1_xreal_0(C,B) | k6_pre_topc(k5_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t21_borsuk_4,t24_topmetr,t19_borsuk_4,t12_pre_topc,t47_pre_topc,t80_topreal6,t28_xboole_1]), [file(borsuk_4,t24_borsuk_4),interesting(0.63)]). fof(t23_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_rcomp_2(B,C) => ( r1_xreal_0(C,B) | k6_pre_topc(k5_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t20_borsuk_4,t24_topmetr,t18_borsuk_4,t12_pre_topc,t47_pre_topc,t80_topreal6,t28_xboole_1]), [file(borsuk_4,t23_borsuk_4),interesting(0.63)]). fof(t8_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => v1_jordan1(k3_topreal1(A,B,C),A) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_jordan1,t12_topreal1]), [file(borsuk_4,t8_borsuk_4),interesting(0.63)]). fof(t43_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) ) ), inference(mizar_proof,[status(thm)],[t83_borsuk_1,t16_rcomp_1]), [file(borsuk_4,t43_borsuk_4),interesting(0.62)]). fof(t76_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ( ( r1_topreal1(k15_euclid(A),D,E,B) & r1_topreal1(k15_euclid(A),E,D,C) & k5_subset_1(u1_struct_0(k15_euclid(A)),B,C) = k2_struct_0(k15_euclid(A),D,E) ) => ( D = E | r1_t_0topsp(k1_borsuk_4,k3_pre_topc(k15_euclid(A),k4_subset_1(u1_struct_0(k15_euclid(A)),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),D)),k6_subset_1(u1_struct_0(k15_euclid(A)),C,k1_struct_0(k15_euclid(A),D))))) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,t2_jordan5a,t74_borsuk_4,t75_borsuk_4,t14_jordan5b,t7_borsuk_4,t7_xboole_1,t3_xboole_1,t1_jordan1,d1_struct_0,t1_jordan1,t1_jordan1,d1_struct_0,d10_pre_topc,d10_pre_topc,t1_jordan1,t1_jordan1,t7_xboole_1,t4_tsep_1,t1_jordan1,t7_xboole_1,t4_tsep_1,t9_funct_2,d5_tops_2,t7_topmetr,t9_funct_2,d5_tops_2,t7_topmetr,d10_pre_topc,d10_pre_topc,t72_borsuk_4,d10_pre_topc,t73_borsuk_4,t40_borsuk_4,d1_borsuk_4,t12_pre_topc,t29_borsuk_4,d1_tarski,t9_jgraph_2,d1_funct_2,t12_pre_topc,d5_tops_2,d5_tops_2,t51_tops_2,t29_borsuk_4,d5_tops_2,d10_pre_topc,d5_tops_2,d10_pre_topc,t49_xboole_1,t49_xboole_1,t41_xboole_1,t23_zfmisc_1,t5_borsuk_4,d4_tops_2,d4_tops_2,d6_partfun1,d1_tarski,t5_circcmb2,d10_pre_topc,d4_tops_2,t6_borsuk_4,t35_borsuk_1,t9_funct_2,t35_borsuk_1,t9_funct_2,t7_xboole_1,t7_xboole_1,t1_jordan1,d10_pre_topc,t1_jordan5a,t16_compts_1,t36_xboole_1,d1_tarski,t4_topreal1,t37_zfmisc_1,t40_zfmisc_1,t49_xboole_1,t23_zfmisc_1,d10_pre_topc,t23_xboole_1,t28_xboole_1,t12_xboole_1,d10_pre_topc,t43_pre_topc,t1_jordan5a,t16_compts_1,t36_xboole_1,d1_tarski,t4_topreal1,t37_zfmisc_1,t40_zfmisc_1,t49_xboole_1,t23_zfmisc_1,d10_pre_topc,t23_xboole_1,t28_xboole_1,t12_xboole_1,d10_pre_topc,t43_pre_topc,d5_tops_2,t7_topmetr,d5_tops_2,t7_topmetr,d1_tarski,t4_rcomp_2,d10_pre_topc,t3_rcomp_2,d10_pre_topc,t56_funct_1,t56_funct_1,t9_jgraph_2,d5_tops_2,d1_t_0topsp]), [file(borsuk_4,t76_borsuk_4),interesting(0.62)]). fof(t82_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ~ ( r1_tarski(A,k2_rcomp_1(0,1)) & ! [B] : ( ( v1_integra1(B) & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) ) => ~ ( r1_tarski(A,B) & r1_tarski(B,k2_rcomp_1(0,1)) & k3_seq_4(A) = k4_pscomp_1(B) & k2_seq_4(A) = k3_pscomp_1(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t83_borsuk_1,t1_xboole_1,t16_compts_1,t52_borsuk_4,t50_borsuk_4,t31_rcomp_1,t30_rcomp_1,d3_tarski,t83_borsuk_1,t51_borsuk_4,t48_rcomp_1,d3_seq_4,t24_seq_4,t49_borsuk_4,t55_borsuk_4,t5_integra1,l85_borsuk_4,t83_borsuk_1,t1_integra2,t47_rcomp_1,t1_integra2,t47_rcomp_1,d3_tarski,t10_borsuk_4,d1_integra1]), [file(borsuk_4,t82_borsuk_4),interesting(0.62)]). fof(t22_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( A = k2_rcomp_1(B,C) => ( r1_xreal_0(C,B) | k6_pre_topc(k5_topmetr,A) = k1_rcomp_1(B,C) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t82_topreal6,t24_topmetr,t17_borsuk_4,t12_pre_topc,t47_pre_topc,t80_topreal6,t28_xboole_1]), [file(borsuk_4,t22_borsuk_4),interesting(0.61)]). fof(t35_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k1_rcomp_1(A,B) = k4_subset_1(k1_numbers,k1_seq_4(A),k2_rcomp_2(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_rcomp_1,t20_rcomp_2]), [file(borsuk_4,t35_borsuk_4),interesting(0.58)]). fof(t34_borsuk_4,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k1_rcomp_1(A,B) = k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_seq_4(B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t14_rcomp_1,t21_rcomp_2]), [file(borsuk_4,t34_borsuk_4),interesting(0.58)]). fof(t54_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k5_topmetr) & v6_compts_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_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,t51_borsuk_4,t48_rcomp_1]), [file(borsuk_4,t54_borsuk_4),interesting(0.56)]). fof(t63_borsuk_4,theorem,( ! [A] : ( ( v1_topreal2(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => r1_borsuk_3(k3_pre_topc(k15_euclid(2),k2_topreal1),k3_pre_topc(k15_euclid(2),A)) ) ), inference(mizar_proof,[status(thm)],[d1_topreal2,d1_t_0topsp]), [file(borsuk_4,t63_borsuk_4),interesting(0.55)]). fof(t31_borsuk_4,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) ) => k6_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_seq_4(B)) = k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k2_rcomp_2(B,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_rcomp_2,t14_rcomp_1,t26_borsuk_4,t70_xboole_1,t4_xboole_1,t88_xboole_1]), [file(borsuk_4,t31_borsuk_4),interesting(0.55)]). fof(t66_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ( B != C => r1_borsuk_3(k5_topmetr,k3_pre_topc(k15_euclid(A),k3_topreal1(A,B,C))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_topreal1,t65_borsuk_4]), [file(borsuk_4,t66_borsuk_4),interesting(0.47)]). fof(t69_borsuk_4,theorem,( ! [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] : ( ( v2_pre_topc(D) & l1_pre_topc(D) ) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ( ( v5_pre_topc(C,A,B) & m1_pre_topc(D,B) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k3_pre_topc(A,E)),u1_struct_0(D)) & m2_relset_1(F,u1_struct_0(k3_pre_topc(A,E)),u1_struct_0(D)) ) => ( F = k2_partfun1(u1_struct_0(A),u1_struct_0(B),C,E) => v5_pre_topc(F,k3_pre_topc(A,E),D) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_jordan1,t38_funct_2,t10_topmetr,t8_topmetr]), [file(borsuk_4,t69_borsuk_4),interesting(0.46)]). fof(t64_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( r1_topreal1(k15_euclid(A),C,D,B) => r1_t_0topsp(k1_borsuk_4,k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k2_struct_0(k15_euclid(A),C,D)))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_topreal1,d5_tops_2,t35_borsuk_1,d1_funct_2,t91_relat_1,t35_borsuk_1,d5_tops_2,t12_pre_topc,t1_jordan1,t2_jordan5a,t58_borsuk_4,t123_funct_1,t146_relat_1,t118_funct_1,t148_relat_1,t1_jordan1,t57_borsuk_4,t38_funct_2,t10_topmetr,t3_funct_2,t12_pre_topc,t36_xboole_1,t56_jordan6,t1_jordan1,t4_goboard9,t12_pre_topc,d5_tops_2,t84_funct_1,t9_topmetr,d5_tops_2,t1_jordan1,t1_jordan1,t38_funct_2,t10_topmetr,t12_pre_topc,d5_tops_2,d4_tops_2,t40_rfunct_2,t9_topmetr,d5_tops_2,d1_t_0topsp]), [file(borsuk_4,t64_borsuk_4),interesting(0.38)]). fof(t68_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k5_topmetr)) => ! [F] : ( m1_subset_1(F,u1_struct_0(k5_topmetr)) => ~ ( r1_topreal1(k15_euclid(A),C,D,B) & ~ r1_xreal_0(F,E) & ! [G] : ( ( ~ v1_xboole_0(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k3_pre_topc(k5_topmetr,G)),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) & m2_relset_1(H,u1_struct_0(k3_pre_topc(k5_topmetr,G)),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) ) => ~ ( G = k1_rcomp_1(E,F) & v3_tops_2(H,k3_pre_topc(k5_topmetr,G),k3_pre_topc(k15_euclid(A),B)) & k1_funct_1(H,E) = C & k1_funct_1(H,F) = D ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_topreal1,t49_borsuk_4,d2_topreal1,t2_jordan5a,t7_jgraph_5,t27_topmetr,t27_topmetr,t20_treal_1,t71_tops_2,d1_treal_1,d2_treal_1,t15_rcomp_1,t1_jordan1,t21_funct_2,t16_treal_1,t15_rcomp_1,t1_jordan1,t21_funct_2,t16_treal_1]), [file(borsuk_4,t68_borsuk_4),interesting(0.21)]). fof(t80_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k3_pre_topc(k15_euclid(2),A)),u1_struct_0(k1_borsuk_4)) & m2_relset_1(B,u1_struct_0(k3_pre_topc(k15_euclid(2),A)),u1_struct_0(k1_borsuk_4)) ) => ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( v3_tops_2(B,k3_pre_topc(k15_euclid(2),A),k1_borsuk_4) & r1_tarski(C,A) & ? [D] : ( m1_subset_1(D,u1_struct_0(k5_topmetr)) & ? [E] : ( m1_subset_1(E,u1_struct_0(k5_topmetr)) & ~ r1_xreal_0(E,D) & k4_pre_topc(k3_pre_topc(k15_euclid(2),A),k1_borsuk_4,B,C) = k1_rcomp_1(D,E) ) ) & ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(2))) => ~ r1_topreal1(k15_euclid(2),D,E,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t70_tops_2,d5_tops_2,d5_tops_2,t152_funct_1,d10_xboole_0,d1_funct_2,t1_jordan1,t146_funct_1,d5_tops_2,t68_tops_2,t148_relat_1,t1_jordan1,t12_pre_topc,t1_jordan1,t38_funct_2,t8_funct_2,d1_funct_2,t91_relat_1,t1_jordan1,t12_pre_topc,t84_funct_1,t1_jordan1,t4_topmetr,d5_tops_2,t69_borsuk_4,d4_tops_2,t40_rfunct_2,d4_tops_2,t65_funct_1,t1_jordan1,t47_jordan6,t69_borsuk_4,d5_tops_2,d1_t_0topsp,t79_borsuk_4]), [file(borsuk_4,t80_borsuk_4),interesting(0.19)]). 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(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(t69_xboole_1,theorem,( ! [A,B] : ( ~ v1_xboole_0(B) => ~ ( r1_tarski(B,A) & r1_xboole_0(B,A) ) ) ), file(xboole_1,t69_xboole_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(t7_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,A) & ~ r1_xreal_0(B,C) ) ) ) ) ) ), file(xreal_1,t7_xreal_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(d2_rcomp_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k2_rcomp_1(A,B) = a_2_1_rcomp_1(A,B) ) ) ), file(rcomp_1,d2_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(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(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(t2_rcomp_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( ~ r1_xreal_0(B,A) & ~ r1_xreal_0(B,C) ) <=> ~ r1_xreal_0(B,k2_square_1(A,C)) ) ) ) ) ), file(rcomp_2,t2_rcomp_2), [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(t13_rcomp_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => k1_rcomp_1(B,A) = k1_xboole_0 ) ) ) ), file(rcomp_1,t13_rcomp_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(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(d2_tarski,definition,( ! [A,B,C] : ( C = k2_tarski(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( D = A | D = B ) ) ) ), file(tarski,d2_tarski), [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(t24_topmetr,theorem,( u1_struct_0(k3_topmetr) = k1_numbers ), file(topmetr,t24_topmetr), [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(t1_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_tarski(k1_rcomp_1(B,C),k1_rcomp_1(A,D)) ) ) ) ) ) ), file(treal_1,t1_treal_1), [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(t47_pre_topc,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 => k6_pre_topc(B,D) = k3_xboole_0(k6_pre_topc(A,C),k2_pre_topc(B)) ) ) ) ) ) ), file(pre_topc,t47_pre_topc), [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(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(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(t16_rcomp_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r2_hidden(A,k1_rcomp_1(B,C)) & r2_hidden(D,k1_rcomp_1(B,C)) ) => r1_tarski(k1_rcomp_1(A,D),k1_rcomp_1(B,C)) ) ) ) ) ) ), file(rcomp_1,t16_rcomp_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(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(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(d1_topreal2,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v1_topreal2(A) <=> ? [B] : ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k3_pre_topc(k15_euclid(2),k2_topreal1)),u1_struct_0(k3_pre_topc(k15_euclid(2),A))) & m2_relset_1(B,u1_struct_0(k3_pre_topc(k15_euclid(2),k2_topreal1)),u1_struct_0(k3_pre_topc(k15_euclid(2),A))) & v3_tops_2(B,k3_pre_topc(k15_euclid(2),k2_topreal1),k3_pre_topc(k15_euclid(2),A)) ) ) ) ), file(topreal2,d1_topreal2), [interesting(0.00)]). fof(d1_t_0topsp,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ( r1_t_0topsp(A,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) ) ) ) ) ), file(t_0topsp,d1_t_0topsp), [interesting(0.00)]). fof(d2_topreal1,definition,( ! [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,k1_zfmisc_1(u1_struct_0(A))) => ( r1_topreal1(A,B,C,D) <=> ? [E] : ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D))) & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(A,D))) & v3_tops_2(E,k5_topmetr,k3_pre_topc(A,D)) & k1_funct_1(E,0) = B & k1_funct_1(E,1) = C ) ) ) ) ) ) ), file(topreal1,d2_topreal1), [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(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(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(t91_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( r1_tarski(A,k1_relat_1(B)) => k1_relat_1(k7_relat_1(B,A)) = A ) ) ), file(relat_1,t91_relat_1), [interesting(0.00)]). fof(t1_jordan1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => u1_struct_0(k3_pre_topc(A,B)) = B ) ) ), file(jordan1,t1_jordan1), [interesting(0.00)]). fof(t2_jordan5a,theorem,( ! [A] : ( v1_xreal_0(A) => ( ( r1_xreal_0(0,A) & r1_xreal_0(A,1) ) <=> r2_hidden(A,u1_struct_0(k5_topmetr)) ) ) ), file(jordan5a,t2_jordan5a), [interesting(0.00)]). fof(d1_borsuk_4,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v1_pre_topc(A) & m1_pre_topc(A,k5_topmetr) ) => ( A = k1_borsuk_4 <=> u1_struct_0(A) = k2_rcomp_1(0,1) ) ) ), file(borsuk_4,d1_borsuk_4), [interesting(0.00)]). fof(t46_rcomp_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => r1_xboole_0(k2_rcomp_1(A,B),k2_tarski(A,B)) ) ) ), file(rcomp_1,t46_rcomp_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(t123_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( v2_funct_1(C) => k9_relat_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ), file(funct_1,t123_funct_1), [interesting(0.00)]). fof(t146_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => k9_relat_1(A,k1_relat_1(A)) = k2_relat_1(A) ) ), file(relat_1,t146_relat_1), [interesting(0.00)]). fof(t118_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( r2_hidden(A,k1_relat_1(C)) & r2_hidden(B,k1_relat_1(C)) ) => k9_relat_1(C,k2_tarski(A,B)) = k2_tarski(k1_funct_1(C,A),k1_funct_1(C,B)) ) ) ), file(funct_1,t118_funct_1), [interesting(0.00)]). fof(t148_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => k2_relat_1(k7_relat_1(B,A)) = k9_relat_1(B,A) ) ), file(relat_1,t148_relat_1), [interesting(0.00)]). fof(t5_tsep_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_pre_topc(C,A) => ( u1_struct_0(B) = u1_struct_0(C) => g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) = g1_pre_topc(u1_struct_0(C),u1_pre_topc(C)) ) ) ) ) ), file(tsep_1,t5_tsep_1), [interesting(0.00)]). fof(t38_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(C,A) => ( ( B = k1_xboole_0 & A != k1_xboole_0 ) | ( v1_funct_1(k2_partfun1(A,B,D,C)) & v1_funct_2(k2_partfun1(A,B,D,C),C,B) & m2_relset_1(k2_partfun1(A,B,D,C),C,B) ) ) ) ) ), file(funct_2,t38_funct_2), [interesting(0.00)]). fof(t10_topmetr,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ! [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)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(k3_pre_topc(A,C)),u1_struct_0(B)) ) => ( ( v5_pre_topc(D,A,B) & E = k2_partfun1(u1_struct_0(A),u1_struct_0(B),D,C) ) => v5_pre_topc(E,k3_pre_topc(A,C),B) ) ) ) ) ) ) ), file(topmetr,t10_topmetr), [interesting(0.00)]). fof(t3_funct_2,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v1_funct_1(A) & v1_funct_2(A,k1_relat_1(A),k2_relat_1(A)) & m2_relset_1(A,k1_relat_1(A),k2_relat_1(A)) ) ) ), file(funct_2,t3_funct_2), [interesting(0.00)]). fof(t36_xboole_1,theorem,( ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ), file(xboole_1,t36_xboole_1), [interesting(0.00)]). fof(t56_jordan6,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ~ ( r1_topreal1(k15_euclid(A),C,D,B) & k6_subset_1(u1_struct_0(k15_euclid(A)),B,k2_struct_0(k15_euclid(A),C,D)) = k1_xboole_0 ) ) ) ) ) ), file(jordan6,t56_jordan6), [interesting(0.00)]). fof(t4_goboard9,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(k3_pre_topc(A,C)))) => ( B = D => k3_pre_topc(A,B) = k3_pre_topc(k3_pre_topc(A,C),D) ) ) ) ) ) ), file(goboard9,t4_goboard9), [interesting(0.00)]). fof(t84_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( v2_funct_1(B) => v2_funct_1(k7_relat_1(B,A)) ) ) ), file(funct_1,t84_funct_1), [interesting(0.00)]). fof(t9_topmetr,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [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(u1_struct_0(B))) => ( v5_pre_topc(C,A,B) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,D))) ) => ( E = C => v5_pre_topc(E,A,k3_pre_topc(B,D)) ) ) ) ) ) ) ) ), file(topmetr,t9_topmetr), [interesting(0.00)]). fof(d4_tops_2,definition,( ! [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)) ) => ( ( k2_relat_1(C) = k2_pre_topc(B) & v2_funct_1(C) ) => k2_tops_2(A,B,C) = k2_funct_1(C) ) ) ) ) ), file(tops_2,d4_tops_2), [interesting(0.00)]). fof(t40_rfunct_2,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) ) => k2_funct_1(k7_relat_1(B,A)) = k7_relat_1(k2_funct_1(B),k9_relat_1(B,A)) ) ), file(rfunct_2,t40_rfunct_2), [interesting(0.00)]). fof(t15_topreal1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ( B != C => r1_topreal1(k15_euclid(A),B,C,k3_topreal1(A,B,C)) ) ) ) ) ), file(topreal1,t15_topreal1), [interesting(0.00)]). fof(t65_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( r1_topreal1(k15_euclid(A),C,D,B) => r1_t_0topsp(k5_topmetr,k3_pre_topc(k15_euclid(A),B)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_topreal1,d1_t_0topsp]), [file(borsuk_4,t65_borsuk_4),interesting(0.00)]). fof(t4_realset1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ( ~ v1_realset1(A) <=> ! [B] : ~ v1_xboole_0(k4_xboole_0(A,k1_tarski(B))) ) ) ), file(realset1,t4_realset1), [interesting(0.00)]). fof(d1_xboole_0,definition,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), [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(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(t5_topreal2,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( v1_topreal2(A) <=> ( ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) & ? [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) & B != C & r2_hidden(B,A) & r2_hidden(C,A) ) ) & ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ~ ( B != C & r2_hidden(B,A) & r2_hidden(C,A) & ! [D] : ( ( ~ v1_xboole_0(D) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [E] : ( ( ~ v1_xboole_0(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( r1_topreal1(k15_euclid(2),B,C,D) & r1_topreal1(k15_euclid(2),B,C,E) & A = k4_subset_1(u1_struct_0(k15_euclid(2)),D,E) & k5_subset_1(u1_struct_0(k15_euclid(2)),D,E) = k2_struct_0(k15_euclid(2),B,C) ) ) ) ) ) ) ) ) ) ), file(topreal2,t5_topreal2), [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(t42_xboole_1,theorem,( ! [A,B,C] : k4_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k4_xboole_0(A,C),k4_xboole_0(B,C)) ), file(xboole_1,t42_xboole_1), [interesting(0.00)]). fof(t14_jordan5b,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( r1_topreal1(k15_euclid(A),C,D,B) => r1_topreal1(k15_euclid(A),D,C,B) ) ) ) ) ) ), file(jordan5b,t14_jordan5b), [interesting(0.00)]). fof(t4_topreal1,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,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ( r1_topreal1(A,C,D,B) => ( r2_hidden(C,B) & r2_hidden(D,B) ) ) ) ) ) ) ), file(topreal1,t4_topreal1), [interesting(0.00)]). fof(t6_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) ) => ( v1_borsuk_1(k4_topmetr(B,C),k4_topmetr(A,D)) & m1_pre_topc(k4_topmetr(B,C),k4_topmetr(A,D)) ) ) ) ) ) ) ), file(treal_1,t6_treal_1), [interesting(0.00)]). fof(d14_borsuk_1,definition,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ( v1_borsuk_1(B,A) <=> ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( C = u1_struct_0(B) => v4_pre_topc(C,A) ) ) ) ) ) ), file(borsuk_1,d14_borsuk_1), [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(t27_topmetr,theorem,( k4_topmetr(0,1) = k22_borsuk_1 ), file(topmetr,t27_topmetr), [interesting(0.00)]). fof(t23_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => v1_connsp_1(k4_topmetr(A,B)) ) ) ) ), file(treal_1,t23_treal_1), [interesting(0.00)]). fof(t7_jgraph_5,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ( ( r1_xreal_0(0,A) & r1_xreal_0(A,B) & r1_xreal_0(B,1) & C = k1_rcomp_1(A,B) ) => k4_topmetr(A,B) = k3_pre_topc(k5_topmetr,C) ) ) ) ) ), file(jgraph_5,t7_jgraph_5), [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(d3_connsp_1,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_connsp_1(B,A) <=> v1_connsp_1(k3_pre_topc(A,B)) ) ) ) ), file(connsp_1,d3_connsp_1), [interesting(0.00)]). fof(t49_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ( r1_xreal_0(A,B) => ( ~ v1_xboole_0(k1_rcomp_1(A,B)) & v2_connsp_1(k1_rcomp_1(A,B),k5_topmetr) & v6_compts_1(k1_rcomp_1(A,B),k5_topmetr) & m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,t83_borsuk_1,t16_rcomp_1,t48_borsuk_4,t23_treal_1,t7_jgraph_5,t17_compts_1,d3_connsp_1,t15_rcomp_1]), [file(borsuk_4,t49_borsuk_4),interesting(0.00)]). fof(t20_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => ( v3_tops_2(k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)),k4_topmetr(0,1),k4_topmetr(A,B)) & k2_tops_2(k4_topmetr(0,1),k4_topmetr(A,B),k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B))) = k4_treal_1(A,B,k1_treal_1(0,1),k2_treal_1(0,1)) & v3_tops_2(k4_treal_1(A,B,k1_treal_1(0,1),k2_treal_1(0,1)),k4_topmetr(A,B),k4_topmetr(0,1)) & k2_tops_2(k4_topmetr(A,B),k4_topmetr(0,1),k4_treal_1(A,B,k1_treal_1(0,1),k2_treal_1(0,1))) = k3_treal_1(A,B,k1_treal_1(A,B),k2_treal_1(A,B)) ) ) ) ) ), file(treal_1,t20_treal_1), [interesting(0.00)]). fof(t71_tops_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(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(B)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(B),u1_struct_0(C)) & m2_relset_1(E,u1_struct_0(B),u1_struct_0(C)) ) => ( ( v3_tops_2(D,A,B) & v3_tops_2(E,B,C) ) => v3_tops_2(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) ) ), file(tops_2,t71_tops_2), [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(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(t21_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_relat_1(E) & v1_funct_1(E) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | k1_funct_1(k5_relat_1(D,E),C) = k1_funct_1(E,k1_funct_1(D,C)) ) ) ) ) ), file(funct_2,t21_funct_2), [interesting(0.00)]). fof(t16_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => ! [C] : ( m1_subset_1(C,u1_struct_0(k4_topmetr(0,1))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k4_topmetr(0,1))) => ( k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(0,1)),k4_treal_1(A,B,C,D),k1_treal_1(A,B)) = C & k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(0,1)),k4_treal_1(A,B,C,D),k2_treal_1(A,B)) = D ) ) ) ) ) ) ), file(treal_1,t16_treal_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(d2_rcomp_2,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k2_rcomp_2(A,B) = a_2_1_rcomp_2(A,B) ) ) ), file(rcomp_2,d2_rcomp_2), [interesting(0.00)]). fof(t61_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ~ ( ~ r1_xreal_0(B,A) & B != 1 & ~ ( ~ v1_xboole_0(k2_rcomp_2(A,B)) & m1_subset_1(k2_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,d5_real_1,d3_tarski,d2_rcomp_2,t2_xreal_1,t2_jordan5a,t60_borsuk_4,t4_rcomp_2]), [file(borsuk_4,t61_borsuk_4),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(t56_zfmisc_1,theorem,( ! [A,B] : ( ~ r2_hidden(A,B) => r1_xboole_0(k1_tarski(A),B) ) ), file(zfmisc_1,t56_zfmisc_1), [interesting(0.00)]). fof(t20_rcomp_2,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) ) => k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k2_rcomp_2(B,C)) = k1_rcomp_1(A,C) ) ) ) ) ), file(rcomp_2,t20_rcomp_2), [interesting(0.00)]). fof(t40_xboole_1,theorem,( ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),B) = k4_xboole_0(A,B) ), file(xboole_1,t40_xboole_1), [interesting(0.00)]). fof(t83_xboole_1,theorem,( ! [A,B] : ( r1_xboole_0(A,B) <=> k4_xboole_0(A,B) = A ) ), file(xboole_1,t83_xboole_1), [interesting(0.00)]). fof(t17_rcomp_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_tarski(k2_rcomp_1(A,B),k1_rcomp_2(A,B)) & r1_tarski(k2_rcomp_1(A,B),k2_rcomp_2(A,B)) & r1_tarski(k1_rcomp_2(A,B),k1_rcomp_1(A,B)) & r1_tarski(k2_rcomp_2(A,B),k1_rcomp_1(A,B)) ) ) ) ), file(rcomp_2,t17_rcomp_2), [interesting(0.00)]). fof(d10_pre_topc,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( ( v1_pre_topc(C) & m1_pre_topc(C,A) ) => ( C = k3_pre_topc(A,B) <=> k2_pre_topc(C) = B ) ) ) ) ), file(pre_topc,d10_pre_topc), [interesting(0.00)]). fof(t117_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r2_hidden(A,k1_relat_1(B)) => k9_relat_1(B,k1_tarski(A)) = k1_tarski(k1_funct_1(B,A)) ) ) ), file(funct_1,t117_funct_1), [interesting(0.00)]). fof(t7_tsep_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_pre_topc(C,B) => m1_pre_topc(C,A) ) ) ) ), file(tsep_1,t7_tsep_1), [interesting(0.00)]). fof(t4_tsep_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_pre_topc(C,A) => ( r1_tarski(u1_struct_0(B),u1_struct_0(C)) <=> m1_pre_topc(B,C) ) ) ) ) ), file(tsep_1,t4_tsep_1), [interesting(0.00)]). fof(t8_topmetr,theorem,( ! [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_pre_topc(D,B) => ( v5_pre_topc(C,A,B) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(D)) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(D)) ) => ( E = C => v5_pre_topc(E,A,D) ) ) ) ) ) ) ) ), file(topmetr,t8_topmetr), [interesting(0.00)]). fof(t47_jordan6,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(k3_pre_topc(A,C)))) => ( ( B = D & r1_tarski(B,C) ) => k3_pre_topc(A,B) = k3_pre_topc(k3_pre_topc(A,C),D) ) ) ) ) ) ), file(jordan6,t47_jordan6), [interesting(0.00)]). fof(t72_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(B,A) => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ), file(funct_1,t72_funct_1), [interesting(0.00)]). fof(t74_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k5_topmetr)) => ! [F] : ( m1_subset_1(F,u1_struct_0(k5_topmetr)) => ~ ( r1_topreal1(k15_euclid(A),C,D,B) & ~ r1_xreal_0(F,E) & F != 1 & ! [G] : ( ( ~ v1_xboole_0(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) ) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k3_pre_topc(k1_borsuk_4,G)),u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),C))))) & m2_relset_1(H,u1_struct_0(k3_pre_topc(k1_borsuk_4,G)),u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),C))))) ) => ~ ( G = k2_rcomp_2(E,F) & v3_tops_2(H,k3_pre_topc(k1_borsuk_4,G),k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),C)))) & k1_funct_1(H,F) = D ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_topreal1,t68_borsuk_4,t61_borsuk_4,t1_jordan1,t27_borsuk_4,t17_rcomp_2,t1_jordan1,d5_tops_2,d5_tops_2,d10_pre_topc,d5_tops_2,d10_pre_topc,t17_rcomp_2,t15_rcomp_1,t123_funct_1,t117_funct_1,t146_relat_1,d10_pre_topc,t148_relat_1,t12_pre_topc,t91_relat_1,d10_pre_topc,t12_pre_topc,t3_funct_2,t1_jordan1,t1_jordan1,t7_tsep_1,t7_tsep_1,t5_tsep_1,t1_jordan1,t1_jordan1,t36_xboole_1,t4_tsep_1,t69_borsuk_4,t84_funct_1,d5_tops_2,t1_jordan1,t4_tsep_1,t36_xboole_1,t36_xboole_1,t47_jordan6,d4_tops_2,t40_rfunct_2,d4_tops_2,d10_pre_topc,t69_borsuk_4,d5_tops_2,t4_rcomp_2,t72_funct_1]), [file(borsuk_4,t74_borsuk_4),interesting(0.00)]). fof(d1_rcomp_2,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k1_rcomp_2(A,B) = a_2_0_rcomp_2(A,B) ) ) ), file(rcomp_2,d1_rcomp_2), [interesting(0.00)]). fof(t62_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k5_topmetr)) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ~ ( ~ r1_xreal_0(B,A) & A != 0 & ~ ( ~ v1_xboole_0(k1_rcomp_2(A,B)) & m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan5a,d3_tarski,d1_rcomp_2,t2_xreal_1,t60_borsuk_4,t3_rcomp_2]), [file(borsuk_4,t62_borsuk_4),interesting(0.00)]). fof(t21_rcomp_2,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) ) => k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_1(B,C)) = k1_rcomp_1(A,C) ) ) ) ) ), file(rcomp_2,t21_rcomp_2), [interesting(0.00)]). fof(t75_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k5_topmetr)) => ! [F] : ( m1_subset_1(F,u1_struct_0(k5_topmetr)) => ~ ( r1_topreal1(k15_euclid(A),C,D,B) & ~ r1_xreal_0(F,E) & E != 0 & ! [G] : ( ( ~ v1_xboole_0(G) & m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) ) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k3_pre_topc(k1_borsuk_4,G)),u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),D))))) & m2_relset_1(H,u1_struct_0(k3_pre_topc(k1_borsuk_4,G)),u1_struct_0(k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),D))))) ) => ~ ( G = k1_rcomp_2(E,F) & v3_tops_2(H,k3_pre_topc(k1_borsuk_4,G),k3_pre_topc(k15_euclid(A),k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),D)))) & k1_funct_1(H,E) = C ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t4_topreal1,t68_borsuk_4,t62_borsuk_4,t1_jordan1,t28_borsuk_4,t17_rcomp_2,t1_jordan1,d5_tops_2,d5_tops_2,d10_pre_topc,d5_tops_2,d10_pre_topc,t17_rcomp_2,t15_rcomp_1,t123_funct_1,t117_funct_1,t146_relat_1,d10_pre_topc,t148_relat_1,t12_pre_topc,t91_relat_1,d10_pre_topc,t12_pre_topc,t3_funct_2,t1_jordan1,t1_jordan1,t7_tsep_1,t7_tsep_1,t5_tsep_1,t1_jordan1,t1_jordan1,t36_xboole_1,t4_tsep_1,t69_borsuk_4,t84_funct_1,d5_tops_2,t1_jordan1,t4_tsep_1,t36_xboole_1,t36_xboole_1,t47_jordan6,d4_tops_2,t40_rfunct_2,d4_tops_2,d10_pre_topc,t69_borsuk_4,d5_tops_2,t3_rcomp_2,t72_funct_1]), [file(borsuk_4,t75_borsuk_4),interesting(0.00)]). fof(t12_zfmisc_1,theorem,( ! [A,B] : r1_tarski(k1_tarski(A),k2_tarski(A,B)) ), file(zfmisc_1,t12_zfmisc_1), [interesting(0.00)]). fof(t34_xboole_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k4_xboole_0(C,B),k4_xboole_0(C,A)) ) ), file(xboole_1,t34_xboole_1), [interesting(0.00)]). fof(t3_xboole_1,theorem,( ! [A] : ( r1_tarski(A,k1_xboole_0) => A = k1_xboole_0 ) ), file(xboole_1,t3_xboole_1), [interesting(0.00)]). fof(t7_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ~ ( r1_topreal1(k15_euclid(A),C,D,B) & v1_xboole_0(k6_subset_1(u1_struct_0(k15_euclid(A)),B,k1_struct_0(k15_euclid(A),C))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t56_jordan6,t12_zfmisc_1,t34_xboole_1,t3_xboole_1]), [file(borsuk_4,t7_borsuk_4),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(d1_struct_0,definition,( ! [A] : ( l1_struct_0(A) => ( v3_struct_0(A) <=> v1_xboole_0(u1_struct_0(A)) ) ) ), file(struct_0,d1_struct_0), [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(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(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(t70_xboole_1,theorem,( ! [A,B,C] : ( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C)) & r1_xboole_0(A,B) & r1_xboole_0(A,C) ) & ~ ( ~ ( r1_xboole_0(A,B) & r1_xboole_0(A,C) ) & r1_xboole_0(A,k2_xboole_0(B,C)) ) ) ), file(xboole_1,t70_xboole_1), [interesting(0.00)]). fof(t12_rcomp_2,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) ) => k4_subset_1(k1_numbers,k2_rcomp_2(A,B),k2_rcomp_2(B,C)) = k2_rcomp_2(A,C) ) ) ) ) ), file(rcomp_2,t12_rcomp_2), [interesting(0.00)]). fof(t6_rcomp_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => k2_rcomp_2(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k1_tarski(B)) ) ) ) ), file(rcomp_2,t6_rcomp_2), [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(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(t36_tops_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))) => ( ( v4_pre_topc(B,A) & v4_pre_topc(C,A) ) => v4_pre_topc(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ), file(tops_1,t36_tops_1), [interesting(0.00)]). fof(t13_rcomp_2,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(A,C) & r1_xreal_0(B,D) & r1_xreal_0(C,D) ) => k1_rcomp_1(A,D) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k1_rcomp_2(A,B),k1_rcomp_1(B,C)),k2_rcomp_2(C,D)) ) ) ) ) ) ), file(rcomp_2,t13_rcomp_2), [interesting(0.00)]). fof(t36_borsuk_4,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(C,B) | k1_rcomp_1(A,D) = k4_subset_1(k1_numbers,k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k2_rcomp_1(B,C)),k1_rcomp_1(C,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_xreal_1,t13_rcomp_2,t35_borsuk_4,t4_xboole_1,t34_borsuk_4,t6_rcomp_2,t4_xboole_1,t4_xboole_1,t35_borsuk_4]), [file(borsuk_4,t36_borsuk_4),interesting(0.00)]). fof(t37_borsuk_4,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(C,B) | k6_subset_1(k1_numbers,k1_rcomp_1(A,D),k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_rcomp_1(C,D))) = k2_rcomp_1(B,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t12_borsuk_4,t13_borsuk_4,t70_xboole_1,t36_borsuk_4,t4_xboole_1,t88_xboole_1]), [file(borsuk_4,t37_borsuk_4),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(t29_tops_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v4_pre_topc(B,A) <=> v3_pre_topc(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_1,t29_tops_1), [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(t5_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => r2_hidden(k1_xboole_0,u1_pre_topc(A)) ) ), file(pre_topc,t5_pre_topc), [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(t41_jordan6,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k22_borsuk_1))) => ( A = k4_xboole_0(u1_struct_0(k22_borsuk_1),k2_tarski(0,1)) => v3_pre_topc(A,k22_borsuk_1) ) ) ), file(jordan6,t41_jordan6), [interesting(0.00)]). fof(d1_tsep_1,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => ( v1_tsep_1(B,A) <=> ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( C = u1_struct_0(B) => v3_pre_topc(C,A) ) ) ) ) ) ), file(tsep_1,d1_tsep_1), [interesting(0.00)]). fof(t17_tsep_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v1_tsep_1(B,A) & 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 => ( v3_pre_topc(D,B) <=> v3_pre_topc(C,A) ) ) ) ) ) ) ), file(tsep_1,t17_tsep_1), [interesting(0.00)]). fof(d6_pre_topc,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v4_pre_topc(B,A) <=> v3_pre_topc(k6_subset_1(u1_struct_0(A),k2_pre_topc(A),B),A) ) ) ) ), file(pre_topc,d6_pre_topc), [interesting(0.00)]). fof(t10_rcomp_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => k2_rcomp_2(B,A) = k1_xboole_0 ) ) ) ), file(rcomp_2,t10_rcomp_2), [interesting(0.00)]). fof(t7_rcomp_2,theorem,( ! [A] : ( v1_xreal_0(A) => k1_rcomp_2(A,A) = k1_xboole_0 ) ), file(rcomp_2,t7_rcomp_2), [interesting(0.00)]). fof(t5_rcomp_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ~ r1_xreal_0(B,A) => k1_rcomp_2(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k1_tarski(A)) ) ) ) ), file(rcomp_2,t5_rcomp_2), [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(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(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(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(t9_jgraph_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)) ) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(D))) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(D))) => ~ ( m1_pre_topc(A,D) & m1_pre_topc(C,D) & G = k2_pre_topc(A) & H = k2_pre_topc(C) & k2_xboole_0(k2_pre_topc(A),k2_pre_topc(C)) = k2_pre_topc(D) & v4_pre_topc(G,D) & v4_pre_topc(H,D) & v5_pre_topc(E,A,B) & v5_pre_topc(F,C,B) & ! [I] : ( r2_hidden(I,k3_xboole_0(k2_pre_topc(A),k2_pre_topc(C))) => k1_funct_1(E,I) = k1_funct_1(F,I) ) & ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,u1_struct_0(D),u1_struct_0(B)) & m2_relset_1(I,u1_struct_0(D),u1_struct_0(B)) ) => ~ ( I = k1_funct_4(E,F) & v5_pre_topc(I,D,B) ) ) ) ) ) ) ) ) ) ) ) ), file(jgraph_2,t9_jgraph_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(t49_xboole_1,theorem,( ! [A,B,C] : k3_xboole_0(A,k4_xboole_0(B,C)) = k4_xboole_0(k3_xboole_0(A,B),C) ), file(xboole_1,t49_xboole_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(t23_zfmisc_1,theorem,( ! [A,B] : ( A != B => k4_xboole_0(k2_tarski(A,B),k1_tarski(B)) = k1_tarski(A) ) ), file(zfmisc_1,t23_zfmisc_1), [interesting(0.00)]). fof(t12_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r2_hidden(A,k1_relat_1(B)) => r2_hidden(k1_funct_1(B,A),k2_relat_1(B)) ) ) ), file(funct_1,t12_funct_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(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(d8_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(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) ) => B = C ) ) ) ), file(funct_1,d8_funct_1), [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(t2_topmetr2,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( v2_funct_1(A) & v2_funct_1(B) & ! [C,D] : ~ ( r2_hidden(C,k1_relat_1(B)) & r2_hidden(D,k4_xboole_0(k1_relat_1(A),k1_relat_1(B))) & k1_funct_1(B,C) = k1_funct_1(A,D) ) ) => v2_funct_1(k1_funct_4(A,B)) ) ) ) ), file(topmetr2,t2_topmetr2), [interesting(0.00)]). fof(t5_borsuk_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v2_funct_1(A) & v2_funct_1(B) & k3_xboole_0(k1_relat_1(A),k1_relat_1(B)) = k1_tarski(C) & k3_xboole_0(k2_relat_1(A),k2_relat_1(B)) = k1_tarski(k1_funct_1(A,C)) ) => v2_funct_1(k1_funct_4(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t12_funct_1,t36_xboole_1,t12_funct_1,t17_xboole_1,t37_zfmisc_1,d3_xboole_0,d1_tarski,d8_funct_1,t3_xboole_0,t79_xboole_1,t2_topmetr2]), [file(borsuk_4,t5_borsuk_4),interesting(0.00)]). fof(d6_partfun1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_partfun1(A,B) <=> ! [C] : ( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B))) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ), file(partfun1,d6_partfun1), [interesting(0.00)]). fof(t5_circcmb2,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_partfun1(A,B) => k2_relat_1(k1_funct_4(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ) ), file(circcmb2,t5_circcmb2), [interesting(0.00)]). fof(t55_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) => ( k2_relat_1(A) = k1_relat_1(k2_funct_1(A)) & k1_relat_1(A) = k2_relat_1(k2_funct_1(A)) ) ) ) ), file(funct_1,t55_funct_1), [interesting(0.00)]). fof(t54_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( B = k2_funct_1(A) <=> ( k1_relat_1(B) = k2_relat_1(A) & ! [C,D] : ( ( ( r2_hidden(C,k2_relat_1(A)) & D = k1_funct_1(B,C) ) => ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) & ( ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) => ( r2_hidden(C,k2_relat_1(A)) & D = k1_funct_1(B,C) ) ) ) ) ) ) ) ) ), file(funct_1,t54_funct_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(t16_funct_4,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( r1_partfun1(B,C) & r2_hidden(A,k1_relat_1(B)) ) => k1_funct_1(k1_funct_4(B,C),A) = k1_funct_1(B,A) ) ) ) ), file(funct_4,t16_funct_4), [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(t13_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(k1_funct_4(B,C))) <=> ( r2_hidden(A,k1_relat_1(B)) | r2_hidden(A,k1_relat_1(C)) ) ) ) ) ), file(funct_4,t13_funct_4), [interesting(0.00)]). fof(t1_jordan5a,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( r1_topreal1(k15_euclid(A),B,C,D) => v6_compts_1(D,k15_euclid(A)) ) ) ) ) ) ), file(jordan5a,t1_jordan5a), [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(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(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(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(t43_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) => ( v4_pre_topc(C,B) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & v4_pre_topc(D,A) & k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) ), file(pre_topc,t43_pre_topc), [interesting(0.00)]). fof(t56_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( ( v2_funct_1(B) & r2_hidden(A,k1_relat_1(B)) ) => ( A = k1_funct_1(k2_funct_1(B),k1_funct_1(B,A)) & A = k1_funct_1(k5_relat_1(B,k2_funct_1(B)),A) ) ) ) ), file(funct_1,t56_funct_1), [interesting(0.00)]). fof(t3_borsuk_3,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) ) => ( ( r1_borsuk_3(A,B) & r1_borsuk_3(B,C) ) => r1_borsuk_3(A,C) ) ) ) ) ), file(borsuk_3,t3_borsuk_3), [interesting(0.00)]). fof(d1_tex_2,definition,( ! [A] : ( ~ v1_xboole_0(A) => ( v1_realset1(A) <=> ? [B] : ( m1_subset_1(B,A) & A = k6_domain_1(A,B) ) ) ) ), file(tex_2,d1_tex_2), [interesting(0.00)]). fof(t15_realset1,theorem,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( ~ ( ~ v1_realset1(B) & ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m1_subset_1(D,A) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & C != D ) ) ) ) & ~ ( ? [C] : ( m1_subset_1(C,A) & ? [D] : ( m1_subset_1(D,A) & r2_hidden(C,B) & r2_hidden(D,B) & C != D ) ) & v1_realset1(B) ) ) ) ), file(realset1,t15_realset1), [interesting(0.00)]). fof(d8_xboole_0,definition,( ! [A,B] : ( r2_xboole_0(A,B) <=> ( r1_tarski(A,B) & A != B ) ) ), file(xboole_0,d8_xboole_0), [interesting(0.00)]). fof(t105_xboole_1,theorem,( ! [A,B] : ~ ( r2_xboole_0(A,B) & k4_xboole_0(B,A) = k1_xboole_0 ) ), file(xboole_1,t105_xboole_1), [interesting(0.00)]). fof(t11_compts_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))) => ( r1_tarski(C,k2_pre_topc(B)) => ( v6_compts_1(C,A) <=> ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) => ( D = C => v6_compts_1(D,B) ) ) ) ) ) ) ) ), file(compts_1,t11_compts_1), [interesting(0.00)]). fof(t24_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(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 => ( v2_connsp_1(C,A) <=> v2_connsp_1(D,B) ) ) ) ) ) ) ), file(connsp_1,t24_connsp_1), [interesting(0.00)]). fof(t24_compts_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [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)) ) => ( ( v5_pre_topc(D,A,C) & k2_relat_1(D) = k2_pre_topc(C) & v6_compts_1(B,A) ) => v6_compts_1(k4_pre_topc(A,C,D,B),C) ) ) ) ) ) ), file(compts_1,t24_compts_1), [interesting(0.00)]). fof(t5_topreal5,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)) & v5_pre_topc(C,A,B) & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( v2_connsp_1(D,A) => v2_connsp_1(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ), file(topreal5,t5_topreal5), [interesting(0.00)]). fof(t39_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) => m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) ) ) ) ), file(pre_topc,t39_pre_topc), [interesting(0.00)]). fof(t43_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] : ( ? [F] : ( r2_hidden(F,A) & r2_hidden(F,C) & E = k1_funct_1(D,F) ) => r2_hidden(E,k9_relat_1(D,C)) ) ) ) ), file(funct_2,t43_funct_2), [interesting(0.00)]). fof(t5_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_borsuk_1(k4_topmetr(A,B),k3_topmetr) & m1_pre_topc(k4_topmetr(A,B),k3_topmetr) ) ) ) ) ), file(treal_1,t5_treal_1), [interesting(0.00)]). fof(t8_tsep_1,theorem,( ! [A] : ( ( v2_pre_topc(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(A))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B))) => ( ( v4_pre_topc(C,A) & r1_tarski(C,u1_struct_0(B)) & r1_tarski(D,C) & D = E ) => ( v4_pre_topc(E,B) <=> v4_pre_topc(D,A) ) ) ) ) ) ) ) ), file(tsep_1,t8_tsep_1), [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(d1_seq_4,definition,( ! [A] : ( v2_membered(A) => ( v1_seq_4(A) <=> ? [B] : ( v1_xreal_0(B) & ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(C,B) ) ) ) ) ) ), file(seq_4,d1_seq_4), [interesting(0.00)]). fof(d2_seq_4,definition,( ! [A] : ( v2_membered(A) => ( v2_seq_4(A) <=> ? [B] : ( v1_xreal_0(B) & ! [C] : ( v1_xreal_0(C) => ( r2_hidden(C,A) => r1_xreal_0(B,C) ) ) ) ) ) ), file(seq_4,d2_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(t32_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( ( v2_rcomp_1(B) & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) ) => ( r1_tarski(A,B) => r1_tarski(k7_pscomp_1(A),B) ) ) ) ), file(pscomp_1,t32_pscomp_1), [interesting(0.00)]). fof(t37_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(k1_numbers)) => ( r1_tarski(A,B) => r1_tarski(k7_pscomp_1(A),k7_pscomp_1(B)) ) ) ) ), file(pscomp_1,t37_pscomp_1), [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(t33_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_topmetr))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ! [E] : ( v1_xreal_0(E) => ( ( A = k1_rcomp_2(C,D) & B = k2_rcomp_2(D,E) ) => ( r1_xreal_0(D,C) | r1_xreal_0(E,D) | r1_connsp_1(k5_topmetr,A,B) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t24_borsuk_4,t30_borsuk_4,d1_connsp_1,t23_borsuk_4,t30_borsuk_4]), [file(borsuk_4,t33_borsuk_4),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(t46_borsuk_4,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v2_connsp_1(A,k5_topmetr) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k5_topmetr))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) => ! [C] : ( m1_subset_1(C,u1_struct_0(k5_topmetr)) => ! [D] : ( m1_subset_1(D,u1_struct_0(k5_topmetr)) => ( ( 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)],[t83_borsuk_1,t40_zfmisc_1,t2_jordan5a,t2_xreal_1,t31_borsuk_4,t17_rcomp_2,t48_rcomp_1,t16_rcomp_1,t83_borsuk_1,t3_rcomp_2,t4_rcomp_2,t1_xboole_1,t33_borsuk_4,t3_rcomp_2,t4_rcomp_2,t17_connsp_1,d5_real_1]), [file(borsuk_4,t46_borsuk_4),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(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(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(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(t1_integra2,theorem,( ! [A] : ( ( v1_integra1(A) & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) ) => ! [B] : ( v1_xreal_0(B) => ( r2_hidden(B,A) <=> ( r1_xreal_0(k4_pscomp_1(A),B) & r1_xreal_0(B,k3_pscomp_1(A)) ) ) ) ) ), file(integra2,t1_integra2), [interesting(0.00)]). fof(t70_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)) ) => ( v3_tops_2(C,A,B) => v3_tops_2(k2_tops_2(A,B,C),B,A) ) ) ) ) ), file(tops_2,t70_tops_2), [interesting(0.00)]). fof(t152_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( v2_funct_1(B) => r1_tarski(k10_relat_1(B,k9_relat_1(B,A)),A) ) ) ), file(funct_1,t152_funct_1), [interesting(0.00)]). fof(t146_funct_1,theorem,( ! [A,B] : ( v1_relat_1(B) => ( r1_tarski(A,k1_relat_1(B)) => r1_tarski(A,k10_relat_1(B,k9_relat_1(B,A))) ) ) ), file(funct_1,t146_funct_1), [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(t8_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(k2_relat_1(D),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,t8_funct_2), [interesting(0.00)]). fof(t4_topmetr,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_pre_topc(C,A) => ( r1_tarski(u1_struct_0(B),u1_struct_0(C)) => m1_pre_topc(B,C) ) ) ) ) ), file(topmetr,t4_topmetr), [interesting(0.00)]). fof(t65_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ( v2_funct_1(A) => k2_funct_1(k2_funct_1(A)) = A ) ) ), file(funct_1,t65_funct_1), [interesting(0.00)]). fof(t67_borsuk_4,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) => ( ? [B] : ( m1_subset_1(B,u1_struct_0(k5_topmetr)) & ? [C] : ( m1_subset_1(C,u1_struct_0(k5_topmetr)) & ~ r1_xreal_0(C,B) & A = k1_rcomp_1(B,C) ) ) => r1_t_0topsp(k5_topmetr,k3_pre_topc(k1_borsuk_4,A)) ) ) ), inference(mizar_proof,[status(thm)],[t7_tsep_1,t2_jordan5a,t27_topmetr,t6_treal_1,t25_topmetr,t1_jordan1,t5_tsep_1,t20_treal_1,t27_topmetr,d1_t_0topsp]), [file(borsuk_4,t67_borsuk_4),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(t79_borsuk_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k1_borsuk_4))) => ~ ( ? [D] : ( m1_subset_1(D,u1_struct_0(k5_topmetr)) & ? [E] : ( m1_subset_1(E,u1_struct_0(k5_topmetr)) & ~ r1_xreal_0(E,D) & C = k1_rcomp_1(D,E) ) ) & r1_t_0topsp(k3_pre_topc(k1_borsuk_4,C),k3_pre_topc(k15_euclid(A),B)) & ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ~ r1_topreal1(k15_euclid(A),D,E,B) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_jordan1,t49_borsuk_4,d1_struct_0,t67_borsuk_4,t3_borsuk_3,d1_t_0topsp,t35_borsuk_1,t2_jordan5a,t7_funct_2,t2_jordan5a,t7_funct_2,d2_topreal1]), [file(borsuk_4,t79_borsuk_4),interesting(0.00)]). fof(t73_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( v6_compts_1(B,k15_euclid(A)) => v1_jordan2c(B,A) ) ) ) ), file(jordan2c,t73_jordan2c), [interesting(0.00)]). fof(t16_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( v1_jordan2c(C,A) & r1_tarski(B,C) ) => v1_jordan2c(B,A) ) ) ) ) ), file(jordan2c,t16_jordan2c), [interesting(0.00)]). fof(t88_topreal6,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ( v1_jordan2c(A,2) & v4_pre_topc(A,k15_euclid(2)) ) => v6_compts_1(A,k15_euclid(2)) ) ) ), file(topreal6,t88_topreal6), [interesting(0.00)]). fof(t20_jordan5a,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( k5_seq_4(k1_rcomp_1(A,B)) = A & k4_seq_4(k1_rcomp_1(A,B)) = B ) ) ) ) ), file(jordan5a,t20_jordan5a), [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(t174_relat_1,theorem,( ! [A,B] : ( v1_relat_1(B) => ~ ( A != k1_xboole_0 & r1_tarski(A,k2_relat_1(B)) & k10_relat_1(B,A) = k1_xboole_0 ) ) ), file(relat_1,t174_relat_1), [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(t147_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(A,k2_relat_1(B)) => k9_relat_1(B,k10_relat_1(B,A)) = A ) ) ), file(funct_1,t147_funct_1), [interesting(0.00)]). fof(t3_jct_misc,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ( r1_tarski(A,k1_relat_1(C)) & r1_tarski(k9_relat_1(C,A),B) ) => r1_tarski(A,k10_relat_1(C,B)) ) ) ), file(jct_misc,t3_jct_misc), [interesting(0.00)]). fof(t4_topreal2,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ ( v1_topreal2(A) & ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ~ ( B != C & r2_hidden(B,A) & r2_hidden(C,A) ) ) ) ) ) ), file(topreal2,t4_topreal2), [interesting(0.00)]). fof(t14_realset1,theorem,( ! [A] : ( ~ ( ~ v1_realset1(A) & ! [B,C] : ~ ( r2_hidden(B,A) & r2_hidden(C,A) & B != C ) ) & ~ ( ? [B,C] : ( r2_hidden(B,A) & r2_hidden(C,A) & B != C ) & v1_realset1(A) ) ) ), file(realset1,t14_realset1), [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(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(t38_zfmisc_1,theorem,( ! [A,B,C] : ( r1_tarski(k2_tarski(A,B),C) <=> ( r2_hidden(A,C) & r2_hidden(B,C) ) ) ), file(zfmisc_1,t38_zfmisc_1), [interesting(0.00)]). fof(t37_xboole_1,theorem,( ! [A,B] : ( k4_xboole_0(A,B) = k1_xboole_0 <=> r1_tarski(A,B) ) ), file(xboole_1,t37_xboole_1), [interesting(0.00)]). fof(t2_xboole_1,theorem,( ! [A] : r1_tarski(k1_xboole_0,A) ), file(xboole_1,t2_xboole_1), [interesting(0.00)]). fof(t60_jordan6,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),A)))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ~ ( v1_topreal2(A) & r2_hidden(C,A) & r2_hidden(D,A) & C != D & B = k6_subset_1(u1_struct_0(k15_euclid(2)),A,k2_struct_0(k15_euclid(2),C,D)) & v2_connsp_1(B,k3_pre_topc(k15_euclid(2),A)) ) ) ) ) ) ), file(jordan6,t60_jordan6), [interesting(0.00)]). fof(d1_jordan1,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( v1_jordan1(B,A) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( ( r2_hidden(C,B) & r2_hidden(D,B) ) => r1_tarski(k3_topreal1(A,C,D),B) ) ) ) ) ) ) ), file(jordan1,d1_jordan1), [interesting(0.00)]). fof(t12_topreal1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ( ( r2_hidden(D,k3_topreal1(A,B,C)) & r2_hidden(E,k3_topreal1(A,B,C)) ) => r1_tarski(k3_topreal1(A,D,E),k3_topreal1(A,B,C)) ) ) ) ) ) ) ), file(topreal1,t12_topreal1), [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(t52_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(k6_xcmplx_0(B,A),0) ) ) ) ), file(xreal_1,t52_xreal_1), [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(t10_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(B,A) & r1_xreal_0(C,D) & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ), file(xreal_1,t10_xreal_1), [interesting(0.00)]). fof(t9_borsuk_4,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(0,D) & r1_xreal_0(D,1) ) => ( r1_xreal_0(C,A) | 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)],[d5_real_1,t70_xreal_1,t52_xreal_1,t66_xreal_1,t10_xreal_1]), [file(borsuk_4,t9_borsuk_4),interesting(0.00)]).