fof(t114_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)))) => ( v1_jordan2c(B,A) => v1_jordan2c(k1_jordan2c(A,B),A) ) ) ) ), inference(mizar_proof,[status(thm)],[t40_jordan2c,t36_xboole_1,t25_euclid,d3_tarski,d4_xboole_0,d4_xboole_0,d5_subset_1,t28_connsp_3,t61_jordan2c,t15_jordan2c,t40_jordan2c,d4_tarski,d3_jordan2c,d6_connsp_1,t25_euclid,d4_xboole_0,d3_xboole_0,d7_xboole_0,t38_connsp_1,d3_jordan2c,t16_jordan2c,t62_jordan2c,t38_nat_1,t1_xreal_1,d3_tarski,d3_tarski,d3_tarski,d4_xboole_0,t24_jordan2b,d2_jordan2b,t12_jordan2c,t29_jordan2b,d2_xboole_0,t29_jordan2b,d2_xboole_0,t9_seq_2,d3_tarski,t24_jordan2b,d2_jordan2b,t132_complex1,t26_xreal_1,t1_jordan2c,t109_real_2,t12_jordan2c,t25_euclid,d4_xboole_0,t24_jordan2b,d2_jordan2b,t132_complex1,d1_absvalue,t12_jordan2c,t25_euclid,d4_xboole_0,d2_xboole_0,d10_xboole_0,t7_xboole_1,t1_xboole_1,t28_connsp_3,t7_xboole_1,t1_xboole_1,t28_connsp_3,t24_jordan2b,d2_jordan2b,t24_jordan2b,d2_jordan2b,d3_tarski,d3_topreal1,t26_jordan2b,t26_jordan2b,t27_jordan2b,t29_jordan2b,t50_xreal_1,t66_xreal_1,t66_xreal_1,t9_xreal_1,d1_jordan1,t14_jordan2c,t24_jordan2b,d2_jordan2b,t24_jordan2b,d2_jordan2b,d3_tarski,d3_topreal1,t26_jordan2b,t26_jordan2b,t27_jordan2b,t29_jordan2b,t50_xreal_1,t66_xreal_1,t66_xreal_1,t9_xreal_1,d1_jordan1,t14_jordan2c,t40_jordan2c,t29_jordan2b,t11_absvalue,t26_xreal_1,t132_complex1,t173_real_2,t26_xreal_1,t2_xreal_1,t12_jordan2c,t138_complex1,t132_complex1,t132_complex1,l5_jordan2c,d1_absvalue,t11_absvalue,t173_real_2,t2_xreal_1,t40_jordan2c,t29_jordan2b,t11_absvalue,t132_complex1,t173_real_2,t2_xreal_1,t12_jordan2c,t132_complex1,t132_complex1,l5_jordan2c,d1_absvalue,t11_absvalue,t173_real_2,t2_xreal_1,t15_jordan2c,t15_jordan2c,t40_jordan2c,d4_tarski,d3_jordan2c,d6_connsp_1,t25_euclid,d4_xboole_0,d3_xboole_0,d7_xboole_0,t38_connsp_1,d3_jordan2c,t16_jordan2c,d3_xboole_0,d7_xboole_0,t38_connsp_1,d3_jordan2c,t16_jordan2c,d2_xboole_0,t38_nat_1,t18_nat_1,t12_pre_topc,t25_euclid,t113_jordan2c,d1_tarski,t24_toprns_1,t40_jordan2c,t16_jordan2c]), [file(jordan2c,t114_jordan2c),interesting(1.00)]). fof(t116_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( ( v1_jordan2c(A,2) & v2_jordan1(A) ) => r1_jordan2c(2,A,k1_jordan2c(2,A)) ) ) ), inference(mizar_proof,[status(thm)],[d2_jordan1,d5_connsp_3,t21_xboole_1,d5_connsp_3,t21_xboole_1,d6_connsp_1,d6_connsp_1,d2_jordan1,t34_connsp_1,t76_jordan2c,d4_jordan2c,d6_connsp_1,d4_jordan2c,t27_jordan2c,t28_xboole_1,d7_xboole_0,d3_jordan2c,t26_jordan2c,d3_tarski,d4_tarski,d3_jordan2c,d6_connsp_1,t34_connsp_1,t37_connsp_1,d7_xboole_0,d3_jordan2c,d4_jordan2c,t37_connsp_1,t7_sprect_1,t28_xboole_1,t23_xboole_1,d7_xboole_0,d10_xboole_0,t28_jordan2c,t31_jordan2c,d10_pre_topc,t12_pre_topc,t115_jordan2c,d3_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,d3_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,d10_xboole_0,t114_jordan2c,d3_jordan2c,t37_connsp_1]), [file(jordan2c,t116_jordan2c),interesting(1.00)]). fof(t59_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r1_xreal_0(2,A) & C = a_2_2_jordan2c(A,B) ) => v2_connsp_1(C,k15_euclid(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_xreal_1,t58_jordan2c,t49_jordan2c,t45_jordan2c,t56_jordan2c,t46_jordan2c,t5_jordan1]), [file(jordan2c,t59_jordan2c),interesting(0.83)]). fof(t76_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)))) => ~ ( r1_xreal_0(2,A) & v1_jordan2c(B,A) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ~ ( r2_jordan2c(A,B,C) & C = k2_jordan2c(A,B) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_jordan2c,t2_xreal_1,d9_tbsp_1,t36_xboole_1,t25_euclid,t13_topreal3,t13_topreal3,t70_jordan2c,t1_jordan1,t1_xboole_1,t3_xboole_0,d4_xboole_0,t13_topreal3,t45_jgraph_1,t13_jordan2c,t4_metric_1,t8_xreal_1,t2_xreal_1,t2_xreal_1,t11_xreal_1,t71_jordan2c,t27_jordan2c,d3_tarski,d4_tarski,t18_jordan2c,t75_jordan2c,t79_xboole_1,d5_connsp_1,d3_tarski,t25_euclid,d4_xboole_0,d3_xboole_0,t13_topreal3,t75_jordan2c,t21_tbsp_1,d3_tarski,t13_topreal3,t52_euclid,t30_toprns_1,t45_jgraph_1,t13_jordan2c,t45_jgraph_1,t8_xreal_1,t2_xreal_1,t31_xreal_1,t2_xreal_1,t13_topreal3,t34_xboole_1,d5_subset_1,d5_subset_1,t43_subset_1,d5_connsp_3,t43_subset_1,t28_xboole_1,d3_connsp_1,t47_jordan6,d3_connsp_1,t28_xboole_1,t26_xboole_1,t26_xboole_1,t3_xboole_1,d5_connsp_3,d7_xboole_0,t74_jordan2c,t16_connsp_3,d10_xboole_0,t13_topreal3,t33_jordan2c,t2_xreal_1,t39_jordan2c,d7_xboole_0,t1_jordan1,t1_xboole_1,t71_jordan2c,t42_jordan2c,t12_pre_topc,t25_euclid,t3_tsep_1,t3_tsep_1,t23_jordan2c,d5_subset_1,t12_pre_topc,t28_connsp_3,t7_connsp_3,t12_pre_topc,t25_euclid]), [file(jordan2c,t76_jordan2c),interesting(0.81)]). fof(t12_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => k5_toprns_1(1,k2_jordan2b(A)) = k18_complex1(A) ) ), inference(mizar_proof,[status(thm)],[t25_euclid,d5_jgraph_1,t116_rvsum_1,d4_square_1,t26_toprns_1,d2_jordan2b,t81_rvsum_1,t103_rvsum_1,t89_square_1,t158_complex1]), [file(jordan2c,t12_jordan2c),interesting(0.79)]). fof(t23_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r3_connsp_1(k15_euclid(A),k2_pre_topc(k15_euclid(A))) ) ), inference(mizar_proof,[status(thm)],[t12_pre_topc,d10_xboole_0,d5_connsp_1]), [file(jordan2c,t23_jordan2c),interesting(0.78)]). fof(t21_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => v1_jordan1(k2_pre_topc(k15_euclid(A)),A) ) ), inference(mizar_proof,[status(thm)],[d1_jordan1,t12_pre_topc]), [file(jordan2c,t21_jordan2c),interesting(0.78)]). fof(t35_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k12_euclid(k4_jordan2c(A)) = k9_square_1(A) ) ), inference(mizar_proof,[status(thm)],[t77_finseq_2,t82_rvsum_1,t59_square_1,t110_rvsum_1]), [file(jordan2c,t35_jordan2c),interesting(0.77)]). fof(t61_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r1_xreal_0(2,A) & C = k4_xboole_0(k1_euclid(A),a_2_3_jordan2c(A,B)) ) => v2_connsp_1(C,k15_euclid(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_xreal_1,t60_jordan2c,t50_jordan2c,t45_jordan2c,t57_jordan2c,t46_jordan2c,t5_jordan1]), [file(jordan2c,t61_jordan2c),interesting(0.77)]). fof(t14_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)))) => ( v1_jordan1(B,A) => v2_connsp_1(B,k15_euclid(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_jordan1,t15_topreal1,d2_topreal1,d5_tops_2,d5_tops_2,d10_pre_topc,d10_pre_topc,t12_pre_topc,t12_pre_topc,t4_topmetr,t83_borsuk_1,d1_funct_2,t4_funct_2,t1_jordan1,t83_borsuk_1,t7_topmetr,t5_jordan1]), [file(jordan2c,t14_jordan2c),interesting(0.77)]). fof(t98_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ v1_jordan2c(k2_goboard9(k1_sprect_1(A)),2) ) ), inference(mizar_proof,[status(thm)],[t40_jordan2c,t26_toprns_1,t26_toprns_1,t31_xreal_1,t31_xreal_1,t32_toprns_1,t56_euclid,t47_jgraph_1,t60_square_1,t89_square_1,t31_xreal_1,t2_xreal_1,t90_sprect_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t33_sprect_1,t34_sprect_1,t91_sprect_1,t92_sprect_1,t93_sprect_1,t94_sprect_1,d1_finseq_6,t55_sprect_2,d3_goboard2,t91_sprect_1,t19_sprect_2,d3_goboard1,t84_pscomp_1,t92_sprect_1,t19_sprect_2,d3_goboard1,t104_pscomp_1,t93_sprect_1,t19_sprect_2,d3_goboard1,t104_pscomp_1,t94_sprect_1,t19_sprect_2,d3_goboard1,t84_pscomp_1,d1_finseq_6,t19_sprect_2,d3_goboard1,t84_pscomp_1,d3_tarski,d5_funct_1,t19_sprect_2,d3_finseq_1,t3_finseq_1,t4_jordan2c,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d3_tarski,t19_sprect_2,d5_funct_1,t19_sprect_2,d5_funct_1,d2_tarski,d10_xboole_0,t20_sprect_2,d4_goboard1,t114_pscomp_1,t20_sprect_2,d4_goboard1,t94_pscomp_1,t20_sprect_2,d4_goboard1,t94_pscomp_1,t20_sprect_2,d4_goboard1,t114_pscomp_1,t20_sprect_2,d4_goboard1,t94_pscomp_1,d3_tarski,d5_funct_1,t20_sprect_2,d3_finseq_1,t3_finseq_1,t4_jordan2c,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d3_tarski,t20_sprect_2,d5_funct_1,t20_sprect_2,d5_funct_1,d2_tarski,d10_xboole_0,t76_card_2,t24_goboard2,t76_card_2,t24_goboard2,t76_card_2,t3_finseq_1,t3_finseq_1,d1_goboard2,d2_goboard2,d4_matrix_1,d1_matrix_1,d3_goboard2,d3_matrix_1,t3_finseq_1,t106_zfmisc_1,t106_zfmisc_1,t25_matrix_1,t97_jordan2c,t97_jordan2c,d3_goboard1,d3_finseq_1,d3_goboard1,d3_goboard1,d5_funct_1,d3_tarski,d2_tarski,t41_enumset1,t80_card_1,d1_tarski,t54_card_2,t79_card_1,t2_xreal_1,t24_goboard2,t31_xreal_1,d7_goboard5,t28_goboard6,t57_euclid,t57_euclid,t60_euclid,t94_pscomp_1,t94_pscomp_1,t62_euclid,t60_euclid,t7_jordan5d,t7_jordan5d,t84_pscomp_1,t104_pscomp_1,t2_xreal_1,t10_xreal_1,t70_xreal_1,t2_xreal_1,t10_xreal_1,t70_xreal_1,t31_xreal_1,d1_goboard9]), [file(jordan2c,t98_jordan2c),interesting(0.77)]). fof(t1_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ( r1_xreal_0(A,0) => k18_complex1(A) = k7_binop_2(A) ) ) ), inference(mizar_proof,[status(thm)],[d1_absvalue,t7_absvalue]), [file(jordan2c,t1_jordan2c),interesting(0.77)]). fof(t42_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(1,A) => k2_jordan2c(A,k1_pre_topc(k15_euclid(A))) = k1_euclid(A) ) ) ), inference(mizar_proof,[status(thm)],[t25_euclid,d10_xboole_0,d3_tarski,t25_euclid,t27_pre_topc,t3_tsep_1,t3_tsep_1,t23_jordan2c,t12_pre_topc,t13_topreal3,d2_jordan2c,t41_jordan2c,t18_jordan2c,t12_pre_topc,d4_tarski]), [file(jordan2c,t42_jordan2c),interesting(0.76)]). fof(t33_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)))) => ( B = k1_euclid(A) => v2_connsp_1(B,k15_euclid(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_euclid,d1_jordan1,t14_jordan2c]), [file(jordan2c,t33_jordan2c),interesting(0.75)]). fof(t41_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(1,A) & v1_jordan2c(k2_pre_topc(k15_euclid(A)),A) ) ) ), inference(mizar_proof,[status(thm)],[d2_jordan2c,t12_pre_topc,t25_euclid,t39_jordan2c]), [file(jordan2c,t41_jordan2c),interesting(0.75)]). fof(t113_jordan2c,theorem,( k1_euclid(0) = k1_struct_0(k15_euclid(0),k16_euclid(0)) ), inference(mizar_proof,[status(thm)],[t112_finseq_2,t72_finseq_2]), [file(jordan2c,t113_jordan2c),interesting(0.75)]). fof(t22_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => v2_connsp_1(k2_pre_topc(k15_euclid(A)),k15_euclid(A)) ) ), inference(mizar_proof,[status(thm)],[t21_jordan2c,t14_jordan2c]), [file(jordan2c,t22_jordan2c),interesting(0.75)]). fof(t28_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)))) => r1_xboole_0(k1_jordan2c(A,B),k2_jordan2c(A,B)) ) ) ), inference(mizar_proof,[status(thm)],[d7_xboole_0,d3_xboole_0,d4_tarski,t17_jordan2c,d4_tarski,t18_jordan2c,d3_xboole_0,d7_xboole_0,t37_connsp_1]), [file(jordan2c,t28_jordan2c),interesting(0.74)]). fof(t63_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(1)))) => ( B = a_1_0_jordan2c(A) => v1_jordan1(B,1) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d3_topreal1,t50_xreal_1,t66_xreal_1,t70_xreal_1,t10_xreal_1,d2_jordan2b,t27_jordan2b,t26_jordan2b,t26_jordan2b,d2_jordan2b,d2_jordan2b,t33_euclid,t33_euclid,t31_euclid,d1_jordan1]), [file(jordan2c,t63_jordan2c),interesting(0.74)]). fof(t64_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(1)))) => ( B = a_1_1_jordan2c(A) => v1_jordan1(B,1) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d3_topreal1,t50_xreal_1,t66_xreal_1,t70_xreal_1,t10_xreal_1,d2_jordan2b,t27_jordan2b,t26_jordan2b,t26_jordan2b,d2_jordan2b,d2_jordan2b,t33_euclid,t33_euclid,t31_euclid,d1_jordan1]), [file(jordan2c,t64_jordan2c),interesting(0.74)]). fof(t58_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( r1_xreal_0(1,A) & a_2_2_jordan2c(A,B) = k1_xboole_0 ) ) ) ), inference(mizar_proof,[status(thm)],[t8_toprns_1,t37_jordan2c,t132_complex1,t94_square_1,t66_xreal_1,t83_square_1,t11_absvalue,t2_xreal_1,t31_xreal_1,t2_xreal_1]), [file(jordan2c,t58_jordan2c),interesting(0.73)]). fof(t79_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( C = a_2_3_jordan2c(A,B) => v2_connsp_1(C,k15_euclid(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t77_jordan2c,t14_jordan2c]), [file(jordan2c,t79_jordan2c),interesting(0.73)]). fof(t37_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k5_toprns_1(A,k5_jordan2c(A)) = k9_square_1(A) ) ), inference(mizar_proof,[status(thm)],[t35_jordan2c,d5_jgraph_1]), [file(jordan2c,t37_jordan2c),interesting(0.71)]). fof(t109_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => v2_tops_1(k3_topreal1(2,A,B),k15_euclid(2)) ) ) ), inference(mizar_proof,[status(thm)],[t7_topreal1,t107_jordan2c,d3_tarski,d3_topreal1,t25_toprns_1,t47_euclid,t26_toprns_1,t13_topreal3,d8_euclid,t22_topmetr,d1_xreal_0,t127_real_2,t56_euclid,t13_topreal3,t50_euclid,t45_euclid,t52_euclid,t27_toprns_1,t8_toprns_1,t47_jgraph_1,t61_square_1,t47_jgraph_1,t153_complex1,d1_absvalue,t88_xcmplx_1,d1_absvalue,t218_xreal_1,t45_jgraph_1,t12_metric_1,d3_topreal1,t50_euclid,t9_jordan2c,t54_euclid,t49_euclid,t54_euclid,t44_euclid,t45_euclid,t53_euclid,t52_euclid,t34_euclid,t52_euclid,t46_euclid,t203_xcmplx_1,t6_xcmplx_1,d9_xcmplx_0,d9_xcmplx_0,t6_xcmplx_1,t35_euclid,t56_euclid,t56_euclid,t58_euclid,t57_euclid,t47_euclid,t58_euclid,t107_xcmplx_1,t33_euclid,t34_euclid,t56_euclid,t108_jordan2c,t47_euclid,d4_xboole_0,d5_subset_1,t3_xboole_0,d13_pre_topc,d4_xboole_0,d5_subset_1,t48_pre_topc,t12_pre_topc,t12_pre_topc,d10_xboole_0,d3_tops_1,d4_tops_1]), [file(jordan2c,t109_jordan2c),interesting(0.71)]). fof(l5_jordan2c,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(0,A) & r1_xreal_0(0,B) ) => r1_xreal_0(0,k2_xcmplx_0(A,B)) ) ) ) ), inference(mizar_proof,[status(thm)],[t35_xreal_1]), [file(jordan2c,l5_jordan2c),interesting(0.71)]). fof(t77_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( C = a_2_3_jordan2c(A,B) => v1_jordan1(C,A) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d3_topreal1,t30_toprns_1,t8_toprns_1,t50_xreal_1,d1_absvalue,t70_xreal_1,t8_toprns_1,t132_complex1,t66_xreal_1,d1_absvalue,t10_xreal_1,t2_xreal_1,t1_xreal_1,t66_xreal_1,t8_toprns_1,d1_absvalue,t70_xreal_1,d1_absvalue,t10_xreal_1,t2_xreal_1,d1_jordan1]), [file(jordan2c,t77_jordan2c),interesting(0.71)]). fof(t96_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_numbers) => ~ ( r1_xreal_0(1,A) & ~ r1_xreal_0(C,0) & B = a_2_6_jordan2c(A,C) & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ~ ( r1_jordan2c(A,B,D) & D = k1_jordan2c(A,B) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t13_topreal3,t13_topreal3,t1_jordan1,t1_xboole_1,d5_connsp_3,d3_tarski,d4_xboole_0,d5_subset_1,t24_toprns_1,d3_xboole_0,t28_xboole_1,t79_jordan2c,d3_connsp_1,t47_jordan6,d3_connsp_1,t9_connsp_3,d5_connsp_1,t13_connsp_3,t1_xboole_1,t13_topreal3,t40_jordan2c,t24_toprns_1,t24_toprns_1,t1_jordan1,t28_xboole_1,d5_connsp_3,t90_jordan2c,t21_connsp_2,t32_tops_2,d10_pre_topc,t88_jordan2c,t34_connsp_3,t38_tops_1,t87_jordan2c,d10_jordan2c,d17_borsuk_1,d10_jordan2c,d18_borsuk_1,t94_jordan2c,d10_jordan2c,t12_pre_topc,t51_tops_2,d5_funct_1,d3_xboole_0,d7_xboole_0,t26_subset_1,d2_jordan2c,t17_jordan2c,d3_tarski,d4_tarski,d3_tarski,d4_tarski,t1_jordan1,t17_jordan2c,d5_subset_1,d4_xboole_0,d5_real_1,d3_tarski,d4_xboole_0,d5_subset_1,t1_jordan1,d3_tarski,t59_jordan2c,d3_connsp_1,t13_topreal3,t59_jordan2c,t69_jordan2c,d6_connsp_3,t3_xboole_0,d7_xboole_0,t32_subset_1,t37_xboole_1,t28_xboole_1,t47_jordan6,d3_connsp_1,d5_connsp_3,t71_jordan2c,t27_jordan2c,t1_connsp_3,t26_jordan2c,d3_xboole_0,t4_xboole_0,t28_jordan2c,t1_connsp_3,d10_xboole_0,t38_nat_1,t1_xreal_1,d3_tarski,d4_tarski,t1_jordan1,t17_jordan2c,d5_subset_1,d4_xboole_0,d5_real_1,t25_euclid,t117_finseq_2,t95_jordan2c,d1_absvalue,d1_absvalue,t26_xreal_1,d3_tarski,d5_jgraph_1,t56_finseq_1,t24_finseq_4,d1_jordan2b,t109_finseq_2,t79_rvsum_1,t57_finseq_1,t158_complex1,t57_finseq_1,t103_rvsum_1,d1_absvalue,d4_xboole_0,d5_subset_1,t1_jordan1,d3_tarski,d3_tarski,t13_topreal3,t13_topreal3,t67_jordan2c,d3_connsp_1,t67_jordan2c,d6_connsp_3,d3_xboole_0,d3_tarski,d5_jgraph_1,t56_finseq_1,t24_finseq_4,d1_jordan2b,t109_finseq_2,t79_rvsum_1,t57_finseq_1,t158_complex1,t57_finseq_1,t103_rvsum_1,d1_absvalue,t26_xboole_1,t3_xboole_1,d7_xboole_0,t3_xboole_1,t32_subset_1,t37_xboole_1,t28_xboole_1,t47_jordan6,d3_connsp_1,d5_connsp_3,t71_jordan2c,t27_jordan2c,t1_connsp_3,t26_jordan2c,d3_xboole_0,d7_xboole_0,t28_jordan2c,d3_tarski,t26_xreal_1,d5_jgraph_1,t56_finseq_1,t24_finseq_4,d1_jordan2b,t109_finseq_2,t79_rvsum_1,t57_finseq_1,t158_complex1,t57_finseq_1,t103_rvsum_1,d1_absvalue,d4_xboole_0,d5_subset_1,t1_jordan1,d3_tarski,d3_tarski,t13_topreal3,t13_topreal3,t68_jordan2c,d3_connsp_1,t68_jordan2c,d6_connsp_3,d3_xboole_0,d3_tarski,t26_xreal_1,t26_xreal_1,d5_jgraph_1,t56_finseq_1,t24_finseq_4,d1_jordan2b,t109_finseq_2,t79_rvsum_1,t57_finseq_1,t158_complex1,t57_finseq_1,t103_rvsum_1,d1_absvalue,t26_xboole_1,t3_xboole_1,d7_xboole_0,t3_xboole_1,t32_subset_1,t37_xboole_1,t28_xboole_1,t47_jordan6,d3_connsp_1,d5_connsp_3,t71_jordan2c,t27_jordan2c,t1_connsp_3,t26_jordan2c,d3_xboole_0,d7_xboole_0,t28_jordan2c,t1_connsp_3,d10_xboole_0]), [file(jordan2c,t96_jordan2c),interesting(0.71)]). fof(t102_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( r1_tarski(k3_goboard9(k1_sprect_1(A)),k1_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)))) & v1_jordan2c(k3_goboard9(k1_sprect_1(A)),2) ) ) ), inference(mizar_proof,[status(thm)],[d2_goboard9,t96_sprect_1,t101_jordan2c,d3_jordan2c,d3_tarski,d4_tarski]), [file(jordan2c,t102_jordan2c),interesting(0.70)]). fof(t69_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r1_xreal_0(2,A) & B = a_2_2_jordan2c(A,C) & D = B ) => ( v2_connsp_1(D,k15_euclid(A)) & ~ v6_tbsp_1(B,k14_euclid(A)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t59_jordan2c,d9_tbsp_1,t8_toprns_1,t37_jordan2c,t132_complex1,t2_xreal_1,t83_square_1,t94_square_1,t66_xreal_1,t11_absvalue,t2_xreal_1,t11_absvalue,t31_xreal_1,t2_xreal_1,t2_xreal_1,t27_toprns_1,t8_toprns_1,t37_jordan2c,t132_complex1,t83_square_1,t94_square_1,t66_xreal_1,t11_absvalue,t2_xreal_1,t2_xreal_1,t132_complex1,t8_xreal_1,t45_jgraph_1,t45_euclid,t39_euclid,t37_euclid,t8_toprns_1,t37_jordan2c,t132_complex1,t66_xreal_1,t11_absvalue,t2_xreal_1,t8_xreal_1,t2_xreal_1,t2_xreal_1]), [file(jordan2c,t69_jordan2c),interesting(0.70)]). fof(t110_jordan2c,theorem,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => v2_tops_1(k5_topreal1(2,A),k15_euclid(2)) ) ), inference(mizar_proof,[status(thm)],[d6_topreal1,d3_tarski,d4_tarski,t38_nat_1,t3_xboole_1,t47_tops_1,t84_tops_1,d3_tarski,d4_tarski,d5_topreal1,d5_topreal1,t47_tops_1,t84_tops_1,t40_sppol_1,d3_tarski,d4_tarski,d4_tarski,t38_nat_1,t1_xreal_1,d2_xboole_0,d10_xboole_0,d3_tarski,d4_tarski,t38_nat_1,d4_tarski,d4_tarski,d5_topreal1,d2_xboole_0,t85_tops_1,s1_nat_1]), [file(jordan2c,t110_jordan2c),interesting(0.68)]). fof(t99_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => r1_tarski(k2_goboard9(k1_sprect_1(A)),k2_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)))) ) ), inference(mizar_proof,[status(thm)],[d1_goboard9,t98_jordan2c,d4_jordan2c,d3_tarski,d4_tarski]), [file(jordan2c,t99_jordan2c),interesting(0.68)]). 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) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_jordan2c,t13_topreal3,t21_tbsp_1,d2_jordan2c]), [file(jordan2c,t16_jordan2c),interesting(0.68)]). fof(t95_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ( C = k13_binarith(k1_numbers,B) => k5_toprns_1(A,C) = k18_complex1(B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_euclid,t109_finseq_2,t56_finseq_1,d5_jgraph_1,t56_finseq_1,t24_finseq_4,d1_jordan2b,t109_finseq_2,t79_rvsum_1,t57_finseq_1,t158_complex1,t57_finseq_1,t103_rvsum_1]), [file(jordan2c,t95_jordan2c),interesting(0.68)]). fof(t39_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ~ ( r1_xreal_0(1,A) & B = k1_euclid(A) & v6_tbsp_1(B,k14_euclid(A)) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_tbsp_1,t13_topreal3,t45_jgraph_1,t13_jordan2c,t8_toprns_1,t37_jordan2c,t31_xreal_1,d1_absvalue,t94_square_1,t66_xreal_1,t83_square_1,t2_xreal_1,t11_xreal_1]), [file(jordan2c,t39_jordan2c),interesting(0.67)]). fof(t13_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k20_euclid(A,B,k16_euclid(A)) = B & k20_euclid(A,k16_euclid(A),B) = k19_euclid(A,B) ) ) ) ), inference(mizar_proof,[status(thm)],[t46_euclid,t51_euclid,t52_euclid,t46_euclid,t45_euclid,t39_euclid,t52_euclid]), [file(jordan2c,t13_jordan2c),interesting(0.66)]). fof(l6_jordan2c,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(0,B) & r1_xreal_0(k2_xcmplx_0(A,B),0) ) ) ) ), inference(mizar_proof,[status(thm)],[t36_xreal_1]), [file(jordan2c,l6_jordan2c),interesting(0.66)]). fof(t24_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)))) => m1_connsp_3(k1_jordan2c(A,B),k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B))) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t17_jordan2c,t17_jordan2c,d2_connsp_3]), [file(jordan2c,t24_jordan2c),interesting(0.66)]). fof(t25_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)))) => m1_connsp_3(k2_jordan2c(A,B),k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B))) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t18_jordan2c,t18_jordan2c,d2_connsp_3]), [file(jordan2c,t25_jordan2c),interesting(0.66)]). fof(t103_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( k2_goboard9(k1_sprect_1(A)) = k2_jordan2c(2,k5_topreal1(2,k1_sprect_1(A))) & k3_goboard9(k1_sprect_1(A)) = k1_jordan2c(2,k5_topreal1(2,k1_sprect_1(A))) ) ) ), inference(mizar_proof,[status(thm)],[t11_gobrd12,t102_jordan2c,t99_jordan2c,d10_xboole_0,d3_tarski,t29_jordan2c,d2_xboole_0,t3_xboole_0,t28_jordan2c,d10_xboole_0,d3_tarski,t30_jordan2c,d2_xboole_0,t3_xboole_0,t28_jordan2c]), [file(jordan2c,t103_jordan2c),interesting(0.66)]). fof(t30_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)))) => r1_tarski(k2_jordan2c(A,B),k3_subset_1(u1_struct_0(k15_euclid(A)),B)) ) ) ), inference(mizar_proof,[status(thm)],[t25_jordan2c,t1_jordan1]), [file(jordan2c,t30_jordan2c),interesting(0.64)]). fof(t36_jordan2c,theorem,( k5_jordan2c(1) = k13_binarith(k1_numbers,1) ), inference(mizar_proof,[status(thm)],[t73_finseq_2]), [file(jordan2c,t36_jordan2c),interesting(0.64)]). fof(t70_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r1_xreal_0(2,A) & B = k4_xboole_0(k1_euclid(A),a_2_3_jordan2c(A,C)) & D = B ) => ( v2_connsp_1(D,k15_euclid(A)) & ~ v6_tbsp_1(B,k14_euclid(A)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t61_jordan2c,t2_xreal_1,t83_square_1,t94_square_1,d9_tbsp_1,t25_euclid,t8_toprns_1,t37_jordan2c,t132_complex1,t66_xreal_1,t11_absvalue,t2_xreal_1,t11_absvalue,t31_xreal_1,t2_xreal_1,t2_xreal_1,d4_xboole_0,t25_euclid,t27_toprns_1,t8_toprns_1,t37_jordan2c,t132_complex1,t66_xreal_1,t11_absvalue,t2_xreal_1,t11_absvalue,t31_xreal_1,t2_xreal_1,t2_xreal_1,d4_xboole_0,t132_complex1,t8_xreal_1,t45_jgraph_1,t45_euclid,t39_euclid,t37_euclid,t8_toprns_1,t37_jordan2c,t132_complex1,t66_xreal_1,t11_absvalue,t2_xreal_1,t8_xreal_1,t2_xreal_1,t2_xreal_1]), [file(jordan2c,t70_jordan2c),interesting(0.63)]). fof(t29_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)))) => r1_tarski(k1_jordan2c(A,B),k3_subset_1(u1_struct_0(k15_euclid(A)),B)) ) ) ), inference(mizar_proof,[status(thm)],[t24_jordan2c,t1_jordan1]), [file(jordan2c,t29_jordan2c),interesting(0.63)]). 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) ) ) ) ), inference(mizar_proof,[status(thm)],[t1_jordan1,d10_pre_topc,t11_compts_1,t10_compts_1,t13_topreal3,t20_topmetr,t12_tbsp_1,t26_tbsp_1,t25_tbsp_1,t12_pre_topc,d2_topmetr,t72_jordan2c,d2_jordan2c,t13_topreal3,d2_jordan2c]), [file(jordan2c,t73_jordan2c),interesting(0.63)]). fof(t27_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)))) => ( r2_jordan2c(A,B,C) => r1_tarski(C,k2_jordan2c(A,B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d4_tarski]), [file(jordan2c,t27_jordan2c),interesting(0.62)]). fof(t65_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(1)))) => ( B = a_1_0_jordan2c(A) => v2_connsp_1(B,k15_euclid(1)) ) ) ) ), inference(mizar_proof,[status(thm)],[t63_jordan2c,t14_jordan2c]), [file(jordan2c,t65_jordan2c),interesting(0.62)]). fof(t66_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(1)))) => ( B = a_1_1_jordan2c(A) => v2_connsp_1(B,k15_euclid(1)) ) ) ) ), inference(mizar_proof,[status(thm)],[t64_jordan2c,t14_jordan2c]), [file(jordan2c,t66_jordan2c),interesting(0.62)]). fof(t38_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(1,A) => r1_xreal_0(1,k5_toprns_1(A,k5_jordan2c(A))) ) ) ), inference(mizar_proof,[status(thm)],[t37_jordan2c,t83_square_1,t94_square_1]), [file(jordan2c,t38_jordan2c),interesting(0.61)]). fof(t104_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( k2_jordan2c(2,k5_topreal1(2,k1_sprect_1(A))) != k1_xboole_0 & r2_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)),k2_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)))) & k1_jordan2c(2,k5_topreal1(2,k1_sprect_1(A))) != k1_xboole_0 & r1_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)),k1_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)))) ) ) ), inference(mizar_proof,[status(thm)],[t103_jordan2c,t103_jordan2c,d1_goboard9,t98_jordan2c,d4_jordan2c,d2_goboard9,t102_jordan2c,d3_jordan2c]), [file(jordan2c,t104_jordan2c),interesting(0.61)]). fof(t26_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)))) => ( r1_jordan2c(A,B,C) => r1_tarski(C,k1_jordan2c(A,B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,d4_tarski]), [file(jordan2c,t26_jordan2c),interesting(0.61)]). fof(t117_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ~ ( ~ r1_xreal_0(B,0) & r2_hidden(C,k5_topreal1(2,k1_sprect_1(A))) & ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ~ ( r2_hidden(D,k1_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)))) & ~ r1_xreal_0(B,k5_toprns_1(2,k20_euclid(2,C,D))) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_sprect_1,t106_jordan2c,d5_connsp_3,t21_xboole_1,d5_connsp_3,t21_xboole_1,t116_jordan2c,d3_jordan2c,d6_connsp_1,t1_jordan1,t104_jordan2c,d3_xboole_0,d7_xboole_0,t37_connsp_1,d4_xboole_0,t13_topreal3,t13_topreal3,d7_metric_1,d3_metric_1,d1_metric_1,t12_metric_1,t6_goboard6,d13_pre_topc,t3_xboole_0,t111_jordan2c,d3_xboole_0,d7_xboole_0,t37_connsp_1,d4_xboole_0,t13_topreal3,t13_topreal3,d7_metric_1,d3_metric_1,d1_metric_1,t12_metric_1,t6_goboard6,d13_pre_topc,t3_xboole_0,t111_jordan2c,d2_xboole_0]), [file(jordan2c,t117_jordan2c),interesting(0.61)]). fof(t34_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k10_euclid(A,k4_jordan2c(A)) = k2_finseq_2(A,1) ) ), inference(mizar_proof,[status(thm)],[t77_finseq_2,t17_finseqop,d2_euclid,d1_absvalue]), [file(jordan2c,t34_jordan2c),interesting(0.61)]). fof(t89_jordan2c,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)))) ) => ( v3_pre_topc(B,k15_euclid(A)) => v1_connsp_2(k3_pre_topc(k15_euclid(A),B)) ) ) ) ), inference(mizar_proof,[status(thm)],[d6_connsp_1,d10_pre_topc,t12_pre_topc,d10_pre_topc,t1_jordan1,t1_xboole_1,t1_xboole_1,t32_tops_2,d10_pre_topc,t38_tops_1,t22_topmetr,d1_xreal_0,d10_pre_topc,t13_topreal3,t1_xboole_1,d10_pre_topc,t12_pre_topc,d3_tarski,t78_jordan2c,t14_jordan2c,t15_jordan2c,t15_jordan2c,t4_goboard6,d3_xboole_0,d7_xboole_0,t38_connsp_1,t22_topmetr,t1_jordan1,t28_xboole_1,t32_tops_2,t24_connsp_2]), [file(jordan2c,t89_jordan2c),interesting(0.61)]). fof(t60_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ~ ( r1_xreal_0(1,A) & k4_xboole_0(k1_euclid(A),a_2_3_jordan2c(A,B)) = k1_xboole_0 ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t25_euclid,d4_xboole_0,t58_jordan2c,t3_xboole_1]), [file(jordan2c,t60_jordan2c),interesting(0.60)]). fof(t101_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(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(k15_euclid(2)))) => ( r4_connsp_1(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),k5_topreal1(2,k1_sprect_1(A))),B) => ( v1_jordan2c(B,2) | B = k2_goboard9(k1_sprect_1(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_goboard9,t98_jordan2c,d6_connsp_1,d6_connsp_1,t40_jordan2c,t36_xboole_1,t25_euclid,t61_jordan2c,d3_tarski,d4_xboole_0,d4_xboole_0,t28_xboole_1,d5_subset_1,d5_connsp_3,t15_jordan2c,t40_jordan2c,t25_euclid,d4_xboole_0,t3_xboole_0,t40_jordan2c,t25_euclid,d4_xboole_0,t3_xboole_0,t100_jordan2c]), [file(jordan2c,t101_jordan2c),interesting(0.58)]). fof(t10_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => k18_complex1(k5_toprns_1(A,B)) = k5_toprns_1(A,B) ) ) ), inference(mizar_proof,[status(thm)],[t26_toprns_1,d1_absvalue]), [file(jordan2c,t10_jordan2c),interesting(0.56)]). fof(t52_jordan2c,theorem,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ( m2_finseq_2(A,k1_numbers,k1_euclid(k3_finseq_1(A))) & m1_subset_1(A,u1_struct_0(k15_euclid(k3_finseq_1(A)))) ) ) ), inference(mizar_proof,[status(thm)],[d11_finseq_1,d4_finseq_2,t25_euclid]), [file(jordan2c,t52_jordan2c),interesting(0.56)]). fof(t112_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(2))) => ~ ( ~ r1_xreal_0(B,0) & r2_hidden(C,k5_topreal1(2,k1_sprect_1(A))) & ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(2))) => ~ ( r2_hidden(D,k2_jordan2c(2,k5_topreal1(2,k1_sprect_1(A)))) & ~ r1_xreal_0(B,k5_toprns_1(2,k20_euclid(2,C,D))) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_sprect_1,t106_jordan2c,d5_connsp_3,t21_xboole_1,d5_connsp_3,t21_xboole_1,t76_jordan2c,d4_jordan2c,d6_connsp_1,t12_pre_topc,d10_pre_topc,t104_jordan2c,d3_xboole_0,d7_xboole_0,t37_connsp_1,d4_xboole_0,t13_topreal3,t13_topreal3,d7_metric_1,d3_metric_1,d1_metric_1,t12_metric_1,t6_goboard6,d13_pre_topc,t3_xboole_0,t111_jordan2c,d3_xboole_0,d7_xboole_0,t37_connsp_1,d4_xboole_0,t13_topreal3,t13_topreal3,d7_metric_1,d3_metric_1,d1_metric_1,t12_metric_1,t6_goboard6,d13_pre_topc,t3_xboole_0,t111_jordan2c,d2_xboole_0]), [file(jordan2c,t112_jordan2c),interesting(0.54)]). fof(t71_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)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ( ( B = E & v2_connsp_1(B,k15_euclid(A)) & C = k1_connsp_3(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),D)),k4_connsp_3(k15_euclid(A),B,k3_subset_1(u1_struct_0(k15_euclid(A)),D))) & r1_xboole_0(E,D) ) => ( v6_tbsp_1(E,k14_euclid(A)) | r2_jordan2c(A,D,C) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d7_xboole_0,t22_subset_1,d4_subset_1,t28_xboole_1,d5_connsp_3,t32_subset_1,t83_xboole_1,t13_topreal3,t43_subset_1,t47_jordan6,d3_connsp_1,d3_connsp_1,t9_connsp_3,t1_connsp_3,t21_tbsp_1,t18_jordan2c]), [file(jordan2c,t71_jordan2c),interesting(0.36)]). fof(t57_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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_xreal_0(2,A) & C = k4_xboole_0(k1_euclid(A),a_2_3_jordan2c(A,B)) & r2_hidden(D,C) & r2_hidden(E,C) & ~ ! [F] : ( m1_subset_1(F,k1_numbers) => ( D != k18_euclid(F,A,E) & E != k18_euclid(F,A,D) ) ) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ! [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(A))) => ! [I] : ( m1_subset_1(I,u1_struct_0(k15_euclid(A))) => ! [J] : ( m1_subset_1(J,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(F,C) & r2_hidden(G,C) & r2_hidden(H,C) & r2_hidden(I,C) & r2_hidden(J,C) & r1_tarski(k3_topreal1(A,D,F),C) & r1_tarski(k3_topreal1(A,F,G),C) & r1_tarski(k3_topreal1(A,G,H),C) & r1_tarski(k3_topreal1(A,H,I),C) & r1_tarski(k3_topreal1(A,I,J),C) & r1_tarski(k3_topreal1(A,J,E),C) ) ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_euclid,t24_toprns_1,d4_xboole_0,t55_jordan2c,t25_euclid,t12_euclid,t11_euclid,d11_euclid,t33_euclid,t127_real_2,d5_jgraph_1,t14_euclid,d1_absvalue,t88_xcmplx_1,d4_xboole_0,d11_euclid,t71_rvsum_1,t71_rvsum_1,d7_xcmplx_0,t74_rvsum_1,t50_jordan2c,t33_euclid,t24_toprns_1,d4_xboole_0,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t34_euclid,t33_euclid,t24_toprns_1,d4_xboole_0,d7_xcmplx_0,t33_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t50_jordan2c,t25_euclid,d3_tarski,t26_toprns_1,d4_xboole_0,d10_xboole_0]), [file(jordan2c,t57_jordan2c),interesting(0.34)]). fof(t44_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,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(C,B) & r2_hidden(D,B) & r2_hidden(E,B) & r1_tarski(k3_topreal1(A,C,D),B) & r1_tarski(k3_topreal1(A,D,E),B) & ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) ) => ~ ( v5_pre_topc(F,k5_topmetr,k3_pre_topc(k15_euclid(A),B)) & C = k1_funct_1(F,0) & E = k1_funct_1(F,1) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t15_topreal1,d2_topreal1,d5_tops_2,d5_tops_2,d10_pre_topc,d10_pre_topc,t12_pre_topc,t12_pre_topc,t4_topmetr,t83_borsuk_1,d1_funct_2,t4_funct_2,t1_jordan1,t83_borsuk_1,t15_topreal1,d2_topreal1,d5_tops_2,d5_tops_2,d10_pre_topc,d10_pre_topc,t12_pre_topc,t12_pre_topc,t4_topmetr,t83_borsuk_1,d1_funct_2,t4_funct_2,t1_jordan1,t83_borsuk_1,t7_topmetr,d10_pre_topc,t7_topmetr,t32_jordan2c,t15_topreal1,d2_topreal1,d5_tops_2,d5_tops_2,d10_pre_topc,d10_pre_topc,t12_pre_topc,t12_pre_topc,t4_topmetr,t83_borsuk_1,d1_funct_2,t4_funct_2,t1_jordan1,t83_borsuk_1,t7_topmetr,t15_topreal1,d2_topreal1,d5_tops_2,d5_tops_2,d10_pre_topc,d10_pre_topc,t12_pre_topc,t12_pre_topc,t4_topmetr,t83_borsuk_1,d1_funct_2,t4_funct_2,t1_jordan1,t83_borsuk_1,t7_topmetr,d10_pre_topc,t4_borsuk_2]), [file(jordan2c,t44_jordan2c),interesting(0.26)]). fof(t87_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,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ~ ( v2_connsp_1(B,k15_euclid(A)) & v3_pre_topc(B,k15_euclid(A)) & r2_hidden(C,B) & r2_hidden(D,B) & C != D & ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(E,k5_topmetr,k15_euclid(A)) & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),E),B) & k1_funct_1(E,0) = C & k1_funct_1(E,1) = D ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t86_jordan2c]), [file(jordan2c,t87_jordan2c),interesting(0.25)]). fof(t45_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,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))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & r2_hidden(E,B) & r2_hidden(F,B) & r1_tarski(k3_topreal1(A,C,D),B) & r1_tarski(k3_topreal1(A,D,E),B) & r1_tarski(k3_topreal1(A,E,F),B) & ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) & m2_relset_1(G,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) ) => ~ ( v5_pre_topc(G,k5_topmetr,k3_pre_topc(k15_euclid(A),B)) & C = k1_funct_1(G,0) & F = k1_funct_1(G,1) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t44_jordan2c,t15_topreal1,d2_topreal1,d5_tops_2,d5_tops_2,d10_pre_topc,d10_pre_topc,t12_pre_topc,t12_pre_topc,t4_topmetr,t83_borsuk_1,d1_funct_2,t4_funct_2,t1_jordan1,t83_borsuk_1,t7_topmetr,d10_pre_topc,t32_jordan2c]), [file(jordan2c,t45_jordan2c),interesting(0.24)]). fof(t106_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k3_subset_1(u1_struct_0(k15_euclid(2)),A) != k1_xboole_0 => ( ( v2_tops_1(A,k15_euclid(2)) & v2_jordan1(A) ) <=> ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & k3_subset_1(u1_struct_0(k15_euclid(2)),A) = k4_subset_1(u1_struct_0(k15_euclid(2)),B,C) & r1_xboole_0(B,C) & k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),B),B) = k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),C),C) & A = k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),B),B) & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ( ( D = B & E = C ) => ( r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),D) & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),E) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_jordan1,t7_xboole_1,t1_jordan1,t7_xboole_1,t1_jordan1,t29_subset_1,t33_xboole_1,d5_subset_1,t33_xboole_1,d5_subset_1,t19_xboole_1,d3_tarski,d4_xboole_0,d5_subset_1,d2_xboole_0,t105_jordan2c,d6_connsp_1,t7_xboole_1,d1_connsp_1,t35_connsp_1,t35_connsp_1,t52_pre_topc,d7_xboole_0,t1_jordan1,t28_xboole_1,t23_xboole_1,t17_xboole_1,t7_xboole_1,t1_xboole_1,t36_connsp_1,t26_xboole_1,d7_xboole_0,t3_xboole_1,d7_xboole_0,d13_pre_topc,t50_pre_topc,d2_xboole_0,d4_xboole_0,d10_xboole_0,d4_xboole_0,d13_pre_topc,t7_xboole_1,t1_jordan1,t7_xboole_1,t1_jordan1,d6_connsp_1,t105_jordan2c,d2_jordan1]), [file(jordan2c,t106_jordan2c),interesting(0.22)]). fof(t48_jordan2c,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(k5_pcomps_1(k14_euclid(A))))) => ~ ( D = k3_topreal1(A,B,C) & ~ r2_hidden(k16_euclid(A),k3_topreal1(A,B,C)) & ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(E,k3_topreal1(A,B,C)) & ~ r1_xreal_0(k5_toprns_1(A,E),0) & k5_toprns_1(A,E) = k1_funct_1(k8_weierstr(k14_euclid(A),D),k16_euclid(A)) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t28_sppol_1,t41_borsuk_1,t36_weierstr,d12_funct_1,d1_tarski,d1_tarski,d1_funct_2,d12_funct_1,d3_weierstr,d1_tarski,d5_weierstr,t22_seq_4,d9_weierstr,d1_tarski,t13_jordan2c,t45_jgraph_1,t26_toprns_1,t25_toprns_1]), [file(jordan2c,t48_jordan2c),interesting(0.21)]). fof(t32_jordan2c,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(E,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) => ~ ( v5_pre_topc(E,k5_topmetr,A) & B = k1_funct_1(E,0) & C = k1_funct_1(E,1) & v5_pre_topc(F,k5_topmetr,A) & C = k1_funct_1(F,0) & D = k1_funct_1(F,1) & ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(G,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) => ~ ( v5_pre_topc(G,k5_topmetr,A) & B = k1_funct_1(G,0) & D = k1_funct_1(G,1) & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(A),G),k4_subset_1(u1_struct_0(A),k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(A),E),k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(A),F))) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_borsuk_2,d2_borsuk_2,d1_borsuk_2,d2_borsuk_2,d1_funct_2,t25_topmetr,d1_funct_2,t25_topmetr,d1_funct_2,t27_topmetr,t46_relat_1,t27_topmetr,d1_funct_4,t46_relat_1,t83_borsuk_1,t2_treal_1,d1_funct_2,t25_topmetr,d1_rcomp_1,d1_funct_2,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t14_treal_1,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t23_funct_1,d1_rcomp_1,t12_funct_4,t45_relat_1,t1_xboole_1,t18_funct_4,t13_xboole_1,t1_xboole_1,d1_funct_2,t11_relset_1,t27_topmetr,t6_treal_1,t15_treal_1,t15_treal_1,t19_funct_2,t27_topmetr,t19_funct_2,t27_topmetr,t19_funct_2,t27_topmetr,t83_borsuk_1,d1_rcomp_1,t27_topmetr,t58_tops_2,t12_pre_topc,t25_topmetr,t12_pre_topc,t25_topmetr,t2_treal_1,t83_borsuk_1,t12_pre_topc,d1_funct_2,t25_topmetr,d1_rcomp_1,d1_funct_2,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t14_treal_1,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t23_funct_1,t1_topmetr2,t38_pcomps_1,d7_topmetr,t11_heine,t3_topmetr,t4_topmetr2,d1_rcomp_1,t14_funct_4,d1_borsuk_2,d2_borsuk_2,d1_rcomp_1,d3_xboole_0,t1_topmetr2,d1_tarski,t12_funct_4,d1_rcomp_1,t25_topmetr,d1_funct_2,t14_funct_4,d1_rcomp_1,t14_funct_4,d3_tarski,d5_funct_1,d1_funct_2,t83_borsuk_1,t83_borsuk_1,t48_rcomp_1,d1_funct_2,d1_funct_2,t121_real_2,t66_xreal_1,t83_borsuk_1,t48_rcomp_1,d5_funct_1,d2_xboole_0,t66_xreal_1,t21_xreal_1,t66_xreal_1,t22_xreal_1,t83_borsuk_1,t48_rcomp_1,d5_funct_1,d2_xboole_0]), [file(jordan2c,t32_jordan2c),interesting(0.20)]). fof(t46_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,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))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ! [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(A))) => ! [I] : ( m1_subset_1(I,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & r2_hidden(E,B) & r2_hidden(F,B) & r2_hidden(G,B) & r2_hidden(H,B) & r2_hidden(I,B) & r1_tarski(k3_topreal1(A,C,D),B) & r1_tarski(k3_topreal1(A,D,E),B) & r1_tarski(k3_topreal1(A,E,F),B) & r1_tarski(k3_topreal1(A,F,G),B) & r1_tarski(k3_topreal1(A,G,H),B) & r1_tarski(k3_topreal1(A,H,I),B) & ! [J] : ( ( v1_funct_1(J) & v1_funct_2(J,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) & m2_relset_1(J,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) ) => ~ ( v5_pre_topc(J,k5_topmetr,k3_pre_topc(k15_euclid(A),B)) & C = k1_funct_1(J,0) & I = k1_funct_1(J,1) ) ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t45_jordan2c,t45_jordan2c,t43_jordan2c]), [file(jordan2c,t46_jordan2c),interesting(0.16)]). fof(t7_topreal1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => k1_topreal1(A,B,B) = k1_struct_0(k15_euclid(A),B) ) ) ), file(topreal1,t7_topreal1), [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(t13_topreal3,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => u1_struct_0(k15_euclid(A)) = u1_struct_0(k14_euclid(A)) ) ), file(topreal3,t13_topreal3), [interesting(0.00)]). fof(t22_topmetr,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ( v3_pre_topc(B,k5_pcomps_1(A)) <=> ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ~ ( r2_hidden(C,B) & ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(D,0) & r1_tarski(k9_metric_1(A,C,D),B) ) ) ) ) ) ) ) ), file(topmetr,t22_topmetr), [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(t93_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k8_square_1(A),0) ) ) ), file(square_1,t93_square_1), [interesting(0.00)]). fof(t127_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) ) | ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(0,B) ) ) & r1_xreal_0(k7_xcmplx_0(A,B),0) ) ) ) ), file(real_2,t127_real_2), [interesting(0.00)]). fof(t51_euclid,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))) => k20_euclid(A,B,k20_euclid(A,C,D)) = k17_euclid(A,k20_euclid(A,B,C),D) ) ) ) ) ), file(euclid,t51_euclid), [interesting(0.00)]). fof(t49_euclid,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))) => k17_euclid(A,B,k20_euclid(A,C,D)) = k20_euclid(A,k17_euclid(A,B,C),D) ) ) ) ) ), file(euclid,t49_euclid), [interesting(0.00)]). fof(t52_euclid,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 = k20_euclid(A,k17_euclid(A,B,C),C) & B = k17_euclid(A,k20_euclid(A,B,C),C) ) ) ) ) ), file(euclid,t52_euclid), [interesting(0.00)]). fof(t8_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k4_real_1(k18_complex1(B),k5_toprns_1(A,C)) = k5_toprns_1(A,k18_euclid(B,A,C)) ) ) ) ), file(toprns_1,t8_toprns_1), [interesting(0.00)]). fof(t77_finseq_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,B) => m2_finseq_1(k2_finseq_2(A,C),B) ) ) ) ), file(finseq_2,t77_finseq_2), [interesting(0.00)]). fof(t82_rvsum_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => k12_rvsum_1(A,k4_finseqop(k1_numbers,A,B)) = k4_finseqop(k1_numbers,A,k7_square_1(B)) ) ) ), file(rvsum_1,t82_rvsum_1), [interesting(0.00)]). fof(t59_square_1,theorem,( k7_square_1(1) = 1 ), file(square_1,t59_square_1), [interesting(0.00)]). fof(t110_rvsum_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => k15_rvsum_1(k4_finseqop(k1_numbers,A,B)) = k11_binop_2(A,B) ) ) ), file(rvsum_1,t110_rvsum_1), [interesting(0.00)]). fof(d5_jgraph_1,definition,( ! [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,k1_numbers) => ( C = k5_toprns_1(A,B) <=> ! [D] : ( m2_finseq_2(D,k1_numbers,k1_euclid(A)) => ( B = D => C = k12_euclid(D) ) ) ) ) ) ) ), file(jgraph_1,d5_jgraph_1), [interesting(0.00)]). fof(t153_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k6_real_1(k17_complex1(A),k17_complex1(B)) = k17_complex1(k7_xcmplx_0(A,B)) ) ) ), file(complex1,t153_complex1), [interesting(0.00)]). fof(d1_absvalue,definition,( ! [A] : ( v1_xreal_0(A) => ( ( r1_xreal_0(0,A) => k16_complex1(A) = A ) & ( ~ r1_xreal_0(0,A) => k16_complex1(A) = k4_xcmplx_0(A) ) ) ) ), file(absvalue,d1_absvalue), [interesting(0.00)]). fof(t88_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ), file(xcmplx_1,t88_xcmplx_1), [interesting(0.00)]). fof(t218_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,k7_xcmplx_0(A,2)) ) ) ), file(xreal_1,t218_xreal_1), [interesting(0.00)]). fof(t45_jgraph_1,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(k14_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k14_euclid(A))) => ( ( D = B & E = C ) => k5_toprns_1(A,k20_euclid(A,B,C)) = k4_metric_1(k14_euclid(A),D,E) ) ) ) ) ) ) ), file(jgraph_1,t45_jgraph_1), [interesting(0.00)]). fof(t12_metric_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( l1_metric_1(B) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => ( r2_hidden(D,k9_metric_1(B,C,A)) <=> ( ~ v3_struct_0(B) & ~ r1_xreal_0(A,k2_metric_1(B,C,D)) ) ) ) ) ) ) ), file(metric_1,t12_metric_1), [interesting(0.00)]). fof(t29_toprns_1,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))) => ( k5_toprns_1(A,k20_euclid(A,B,C)) = 0 <=> B = C ) ) ) ) ), file(toprns_1,t29_toprns_1), [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(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(d5_subset_1,definition,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => k3_subset_1(A,B) = k4_xboole_0(A,B) ) ), file(subset_1,d5_subset_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(d13_pre_topc,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))) => ( C = k6_pre_topc(A,B) <=> ! [D] : ( r2_hidden(D,u1_struct_0(A)) => ( r2_hidden(D,C) <=> ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( v3_pre_topc(E,A) & r2_hidden(D,E) & r1_xboole_0(B,E) ) ) ) ) ) ) ) ) ), file(pre_topc,d13_pre_topc), [interesting(0.00)]). fof(t48_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => r1_tarski(B,k6_pre_topc(A,B)) ) ) ), file(pre_topc,t48_pre_topc), [interesting(0.00)]). fof(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(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(d3_tops_1,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v1_tops_1(B,A) <=> k6_pre_topc(A,B) = k2_pre_topc(A) ) ) ) ), file(tops_1,d3_tops_1), [interesting(0.00)]). fof(d4_tops_1,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_tops_1(B,A) <=> v1_tops_1(k3_subset_1(u1_struct_0(A),B),A) ) ) ) ), file(tops_1,d4_tops_1), [interesting(0.00)]). fof(t107_jordan2c,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r1_xreal_0(1,A) & C = k1_struct_0(k15_euclid(A),B) ) => v2_tops_1(C,k15_euclid(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t13_topreal3,t22_topmetr,d1_xreal_0,t93_square_1,t127_real_2,t13_topreal3,t51_euclid,t49_euclid,t52_euclid,t8_toprns_1,t37_jordan2c,t153_complex1,d1_absvalue,t88_xcmplx_1,d1_absvalue,t218_xreal_1,t45_jgraph_1,t12_metric_1,t29_toprns_1,d1_tarski,d4_xboole_0,d5_subset_1,t3_xboole_0,d13_pre_topc,d1_tarski,d4_xboole_0,d5_subset_1,t48_pre_topc,t12_pre_topc,t12_pre_topc,d10_xboole_0,d3_tops_1,d4_tops_1]), [file(jordan2c,t107_jordan2c),interesting(0.00)]). fof(d3_topreal1,definition,( ! [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))) => k1_topreal1(A,B,C) = a_3_0_topreal1(A,B,C) ) ) ) ), file(topreal1,d3_topreal1), [interesting(0.00)]). fof(t25_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k5_toprns_1(A,B) = 0 => B = k16_euclid(A) ) ) ) ), file(toprns_1,t25_toprns_1), [interesting(0.00)]). fof(t47_euclid,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))) => ( k20_euclid(A,B,C) = k16_euclid(A) => B = C ) ) ) ) ), file(euclid,t47_euclid), [interesting(0.00)]). fof(t26_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => r1_xreal_0(0,k5_toprns_1(A,B)) ) ) ), file(toprns_1,t26_toprns_1), [interesting(0.00)]). fof(d8_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k15_euclid(A) = k5_pcomps_1(k14_euclid(A)) ) ), file(euclid,d8_euclid), [interesting(0.00)]). fof(t56_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( k21_euclid(k23_euclid(A,B)) = A & k22_euclid(k23_euclid(A,B)) = B ) ) ) ), file(euclid,t56_euclid), [interesting(0.00)]). fof(t50_euclid,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))) => k20_euclid(A,B,k17_euclid(A,C,D)) = k20_euclid(A,k20_euclid(A,B,C),D) ) ) ) ) ), file(euclid,t50_euclid), [interesting(0.00)]). fof(t45_euclid,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))) => k20_euclid(A,B,C) = k17_euclid(A,B,k19_euclid(A,C)) ) ) ) ), file(euclid,t45_euclid), [interesting(0.00)]). fof(t27_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => k5_toprns_1(A,k19_euclid(A,B)) = k5_toprns_1(A,B) ) ) ), file(toprns_1,t27_toprns_1), [interesting(0.00)]). fof(t47_jgraph_1,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => k5_toprns_1(2,A) = k9_square_1(k3_real_1(k7_square_1(k21_euclid(A)),k7_square_1(k22_euclid(A)))) ) ), file(jgraph_1,t47_jgraph_1), [interesting(0.00)]). fof(t61_square_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k5_square_1(A) = k5_square_1(k4_xcmplx_0(A)) ) ), file(square_1,t61_square_1), [interesting(0.00)]). fof(t30_euclid,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))) => k17_euclid(A,k17_euclid(A,B,C),D) = k17_euclid(A,B,k17_euclid(A,C,D)) ) ) ) ) ), file(euclid,t30_euclid), [interesting(0.00)]). fof(t9_jordan2c,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))) => k20_euclid(A,k17_euclid(A,B,C),D) = k17_euclid(A,k20_euclid(A,B,D),C) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t45_euclid,t30_euclid,t45_euclid]), [file(jordan2c,t9_jordan2c),interesting(0.00)]). fof(t54_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k18_euclid(k6_xcmplx_0(C,D),A,B) = k20_euclid(A,k18_euclid(C,A,B),k18_euclid(D,A,B)) ) ) ) ) ), file(euclid,t54_euclid), [interesting(0.00)]). fof(t44_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ( k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(k4_xcmplx_0(C),A,B) & k19_euclid(A,k18_euclid(C,A,B)) = k18_euclid(C,A,k19_euclid(A,B)) ) ) ) ) ), file(euclid,t44_euclid), [interesting(0.00)]). fof(t53_euclid,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] : ( v1_xreal_0(D) => k18_euclid(D,A,k20_euclid(A,B,C)) = k20_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ), file(euclid,t53_euclid), [interesting(0.00)]). fof(t34_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k18_euclid(k3_xcmplx_0(C,D),A,B) = k18_euclid(C,A,k18_euclid(D,A,B)) ) ) ) ) ), file(euclid,t34_euclid), [interesting(0.00)]). fof(t46_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => k20_euclid(A,B,B) = k16_euclid(A) ) ) ), file(euclid,t46_euclid), [interesting(0.00)]). fof(t203_xcmplx_1,theorem,( k5_xcmplx_0(0) = 0 ), file(xcmplx_1,t203_xcmplx_1), [interesting(0.00)]). fof(t6_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ~ ( k3_xcmplx_0(A,B) = 0 & A != 0 & B != 0 ) ) ) ), file(xcmplx_1,t6_xcmplx_1), [interesting(0.00)]). fof(d9_xcmplx_0,definition,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ), file(xcmplx_0,d9_xcmplx_0), [interesting(0.00)]). fof(t35_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ~ ( k18_euclid(C,A,B) = k16_euclid(A) & C != 0 & B != k16_euclid(A) ) ) ) ) ), file(euclid,t35_euclid), [interesting(0.00)]). fof(t58_euclid,theorem,( k16_euclid(2) = k23_euclid(0,0) ), file(euclid,t58_euclid), [interesting(0.00)]). fof(t57_euclid,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => A = k23_euclid(k21_euclid(A),k22_euclid(A)) ) ), file(euclid,t57_euclid), [interesting(0.00)]). fof(t107_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ( A != 0 => k3_xcmplx_0(A,k7_xcmplx_0(1,A)) = 1 ) ) ), file(xcmplx_1,t107_xcmplx_1), [interesting(0.00)]). fof(t33_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k18_euclid(1,A,B) = B & k18_euclid(0,A,B) = k16_euclid(A) ) ) ) ), file(euclid,t33_euclid), [interesting(0.00)]). fof(t61_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => k18_euclid(A,2,B) = k23_euclid(k3_xcmplx_0(A,k21_euclid(B)),k3_xcmplx_0(A,k22_euclid(B))) ) ) ), file(euclid,t61_euclid), [interesting(0.00)]). fof(t175_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k3_xcmplx_0(k4_xcmplx_0(A),B) = k4_xcmplx_0(k3_xcmplx_0(A,B)) ) ) ), file(xcmplx_1,t175_xcmplx_1), [interesting(0.00)]). fof(t65_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => r1_xreal_0(0,k3_xcmplx_0(A,A)) ) ), file(xreal_1,t65_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(t108_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ! [C] : ( m1_subset_1(C,k1_numbers) => ( ( k21_euclid(A) = k22_euclid(B) & k7_binop_2(k22_euclid(A)) = k21_euclid(B) & A = k18_euclid(C,2,B) ) => ( k21_euclid(A) = 0 & k22_euclid(A) = 0 & A = k16_euclid(2) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t61_euclid,t56_euclid,t175_xcmplx_1,t65_xreal_1,t10_xreal_1,t6_xcmplx_1,t6_xcmplx_1,t57_euclid,t58_euclid]), [file(jordan2c,t108_jordan2c),interesting(0.00)]). fof(d6_topreal1,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(A))) => k5_topreal1(A,B) = k3_tarski(a_2_0_topreal1(A,B)) ) ) ), file(topreal1,d6_topreal1), [interesting(0.00)]). fof(d4_tarski,definition,( ! [A,B] : ( B = k3_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(C,D) & r2_hidden(D,A) ) ) ) ), file(tarski,d4_tarski), [interesting(0.00)]). fof(t38_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_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(t47_tops_1,theorem,( ! [A] : ( l1_pre_topc(A) => k1_tops_1(A,k1_pre_topc(A)) = k1_pre_topc(A) ) ), file(tops_1,t47_tops_1), [interesting(0.00)]). fof(t84_tops_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_tops_1(B,A) <=> k1_tops_1(A,B) = k1_xboole_0 ) ) ) ), file(tops_1,t84_tops_1), [interesting(0.00)]). fof(d5_topreal1,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( ( r1_xreal_0(1,C) & r1_xreal_0(k1_nat_1(C,1),k3_finseq_1(B)) ) => k4_topreal1(A,B,C) = k3_topreal1(A,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,C),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(A)),B,k1_nat_1(C,1))) ) & ( ~ ( r1_xreal_0(1,C) & r1_xreal_0(k1_nat_1(C,1),k3_finseq_1(B)) ) => k4_topreal1(A,B,C) = k1_xboole_0 ) ) ) ) ) ), file(topreal1,d5_topreal1), [interesting(0.00)]). fof(t40_sppol_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,u1_struct_0(k15_euclid(2))) => v4_pre_topc(k4_topreal1(2,B,A),k15_euclid(2)) ) ) ), file(sppol_1,t40_sppol_1), [interesting(0.00)]). fof(t1_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,A) ) => A = B ) ) ) ), file(xreal_1,t1_xreal_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(t85_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))) => ( ( v2_tops_1(B,A) & v2_tops_1(C,A) & v4_pre_topc(C,A) ) => v2_tops_1(k4_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ), file(tops_1,t85_tops_1), [interesting(0.00)]). fof(s1_nat_1,theorem, ( ( p1_s1_nat_1(0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( p1_s1_nat_1(A) => p1_s1_nat_1(k1_nat_1(A,1)) ) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => p1_s1_nat_1(A) ) ), file(nat_1,s1_nat_1), [interesting(0.00)]). fof(d3_sprect_1,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v2_jordan1(A) <=> ( k3_subset_1(u1_struct_0(k15_euclid(2)),A) != k1_xboole_0 & ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & k3_subset_1(u1_struct_0(k15_euclid(2)),A) = k4_subset_1(u1_struct_0(k15_euclid(2)),B,C) & r1_xboole_0(B,C) & k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),B),B) = k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),C),C) & r4_connsp_1(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A),B) & r4_connsp_1(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A),C) ) ) ) ) ) ), file(sprect_1,d3_sprect_1), [interesting(0.00)]). fof(d2_jordan1,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( v2_jordan1(A) <=> ( k3_subset_1(u1_struct_0(k15_euclid(2)),A) != k1_xboole_0 & ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) & k3_subset_1(u1_struct_0(k15_euclid(2)),A) = k4_subset_1(u1_struct_0(k15_euclid(2)),B,C) & r1_xboole_0(B,C) & k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),B),B) = k6_subset_1(u1_struct_0(k15_euclid(2)),k6_pre_topc(k15_euclid(2),C),C) & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A))))) => ( ( D = B & E = C ) => ( r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),D) & r3_connsp_1(k3_pre_topc(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),A)),E) ) ) ) ) ) ) ) ) ) ), file(jordan1,d2_jordan1), [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(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(t29_subset_1,theorem,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => k3_subset_1(A,k4_subset_1(A,B,C)) = k5_subset_1(A,k3_subset_1(A,B),k3_subset_1(A,C)) ) ) ), file(subset_1,t29_subset_1), [interesting(0.00)]). fof(t33_xboole_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k4_xboole_0(A,C),k4_xboole_0(B,C)) ) ), file(xboole_1,t33_xboole_1), [interesting(0.00)]). fof(t19_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(A,C) ) => r1_tarski(A,k3_xboole_0(B,C)) ) ), file(xboole_1,t19_xboole_1), [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(t43_connsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r3_connsp_1(A,B) <=> ? [C] : ( m1_subset_1(C,u1_struct_0(A)) & B = k1_connsp_1(A,C) ) ) ) ) ), file(connsp_1,t43_connsp_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(d6_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))) => ( r4_connsp_1(A,B,C) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(A,B)))) & D = C & r3_connsp_1(k3_pre_topc(A,B),D) ) ) ) ) ) ), file(connsp_1,d6_connsp_1), [interesting(0.00)]). fof(t40_connsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => r2_hidden(B,k1_connsp_1(A,B)) ) ) ), file(connsp_1,t40_connsp_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(t4_xboole_0,theorem,( ! [A,B] : ( ~ ( ~ r1_xboole_0(A,B) & ! [C] : ~ r2_hidden(C,k3_xboole_0(A,B)) ) & ~ ( ? [C] : r2_hidden(C,k3_xboole_0(A,B)) & r1_xboole_0(A,B) ) ) ), file(xboole_0,t4_xboole_0), [interesting(0.00)]). fof(t26_xboole_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k3_xboole_0(A,C),k3_xboole_0(B,C)) ) ), file(xboole_1,t26_xboole_1), [interesting(0.00)]). fof(d7_xboole_0,definition,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), [interesting(0.00)]). fof(t105_jordan2c,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( k3_subset_1(u1_struct_0(A),B) != k1_xboole_0 => ( v2_tops_1(B,A) <=> ! [C,D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r2_hidden(C,B) & r2_hidden(C,D) & v3_pre_topc(D,A) & ! [E] : ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r4_connsp_1(A,k3_subset_1(u1_struct_0(A),B),E) & ~ r1_xboole_0(D,E) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_tops_1,d3_tops_1,t12_pre_topc,d13_pre_topc,t3_xboole_0,t17_pre_topc,t12_pre_topc,d5_subset_1,t1_jordan1,t43_connsp_1,t1_jordan1,t1_xboole_1,d6_connsp_1,t40_connsp_1,d3_xboole_0,t4_xboole_0,d3_tarski,d6_connsp_1,t1_jordan1,t26_xboole_1,d7_xboole_0,t3_xboole_1,d7_xboole_0,d13_pre_topc,d4_xboole_0,d5_subset_1,t48_pre_topc,t12_pre_topc,t12_pre_topc,d10_xboole_0,d3_tops_1,d4_tops_1]), [file(jordan2c,t105_jordan2c),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(t35_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r3_connsp_1(A,B) => v4_pre_topc(B,A) ) ) ) ), file(connsp_1,t35_connsp_1), [interesting(0.00)]). fof(t52_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ( v4_pre_topc(B,A) => k6_pre_topc(A,B) = B ) & ( ( v2_pre_topc(A) & k6_pre_topc(A,B) = B ) => v4_pre_topc(B,A) ) ) ) ) ), file(pre_topc,t52_pre_topc), [interesting(0.00)]). fof(t28_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), [interesting(0.00)]). fof(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(t17_xboole_1,theorem,( ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ), file(xboole_1,t17_xboole_1), [interesting(0.00)]). fof(t36_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))) => ~ ( r3_connsp_1(A,B) & r3_connsp_1(A,C) & B != C & ~ r1_connsp_1(A,B,C) ) ) ) ) ), file(connsp_1,t36_connsp_1), [interesting(0.00)]). fof(t50_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => k6_pre_topc(A,k4_subset_1(u1_struct_0(A),B,C)) = k4_subset_1(u1_struct_0(A),k6_pre_topc(A,B),k6_pre_topc(A,C)) ) ) ) ), file(pre_topc,t50_pre_topc), [interesting(0.00)]). fof(d5_connsp_3,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))) => k4_connsp_3(A,B,C) = k5_subset_1(u1_struct_0(A),B,C) ) ) ) ), file(connsp_3,d5_connsp_3), [interesting(0.00)]). fof(t21_xboole_1,theorem,( ! [A,B] : k3_xboole_0(A,k2_xboole_0(A,B)) = A ), file(xboole_1,t21_xboole_1), [interesting(0.00)]). fof(d2_jordan2c,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_jordan2c(B,A) <=> ( v6_tbsp_1(B,k14_euclid(A)) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) ) ) ) ), file(jordan2c,d2_jordan2c), [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(d9_tbsp_1,definition,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v6_tbsp_1(B,A) <=> ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ( ( r2_hidden(D,B) & r2_hidden(E,B) ) => r1_xreal_0(k2_metric_1(A,D,E),C) ) ) ) ) ) ) ) ), file(tbsp_1,d9_tbsp_1), [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(t25_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => u1_struct_0(k15_euclid(A)) = k1_euclid(A) ) ), file(euclid,t25_euclid), [interesting(0.00)]). fof(t132_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => r1_xreal_0(0,k17_complex1(A)) ) ), file(complex1,t132_complex1), [interesting(0.00)]). fof(t94_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(0,A) & r1_xreal_0(A,B) ) => r1_xreal_0(k8_square_1(A),k8_square_1(B)) ) ) ) ), file(square_1,t94_square_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(t83_square_1,theorem,( k9_square_1(1) = 1 ), file(square_1,t83_square_1), [interesting(0.00)]). fof(t11_absvalue,theorem,( ! [A] : ( v1_xreal_0(A) => ( r1_xreal_0(k1_real_1(k18_complex1(A)),A) & r1_xreal_0(A,k18_complex1(A)) ) ) ), file(absvalue,t11_absvalue), [interesting(0.00)]). fof(t31_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,0) & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ), file(xreal_1,t31_xreal_1), [interesting(0.00)]). fof(t39_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => k19_euclid(A,k19_euclid(A,B)) = B ) ) ), file(euclid,t39_euclid), [interesting(0.00)]). fof(d7_xcmplx_0,definition,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( ( A != 0 => ( B = k5_xcmplx_0(A) <=> k3_xcmplx_0(A,B) = 1 ) ) & ( A = 0 => ( B = k5_xcmplx_0(A) <=> B = 0 ) ) ) ) ) ), file(xcmplx_0,d7_xcmplx_0), [interesting(0.00)]). fof(t47_jordan2c,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_numbers) => ( B != k18_euclid(D,A,C) & C != k18_euclid(D,A,B) ) ) & r2_hidden(k16_euclid(A),k3_topreal1(A,B,C)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_topreal1,t52_euclid,t13_jordan2c,t44_euclid,t34_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t33_euclid]), [file(jordan2c,t47_jordan2c),interesting(0.00)]). fof(t28_sppol_1,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))) => v6_compts_1(k3_topreal1(A,B,C),k15_euclid(A)) ) ) ) ), file(sppol_1,t28_sppol_1), [interesting(0.00)]). fof(t41_borsuk_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => v6_compts_1(k1_struct_0(A,B),A) ) ) ), file(borsuk_1,t41_borsuk_1), [interesting(0.00)]). fof(t36_weierstr,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ~ ( B != k1_xboole_0 & v6_compts_1(B,k5_pcomps_1(A)) & C != k1_xboole_0 & v6_compts_1(C,k5_pcomps_1(A)) & ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ~ ( r2_hidden(D,B) & r2_hidden(E,C) & k4_metric_1(A,D,E) = k9_weierstr(A,B,C) ) ) ) ) ) ) ) ), file(weierstr,t36_weierstr), [interesting(0.00)]). fof(d12_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k9_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E] : ( r2_hidden(E,k1_relat_1(A)) & r2_hidden(E,B) & D = k1_funct_1(A,E) ) ) ) ) ), file(funct_1,d12_funct_1), [interesting(0.00)]). fof(d1_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(d3_weierstr,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => k3_weierstr(A) = A ) ), file(weierstr,d3_weierstr), [interesting(0.00)]). fof(d5_weierstr,definition,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr))) => k5_weierstr(A) = k5_seq_4(k3_weierstr(A)) ) ), file(weierstr,d5_weierstr), [interesting(0.00)]). fof(t22_seq_4,theorem,( ! [A] : ( v1_xreal_0(A) => ( k5_seq_4(k1_seq_4(A)) = A & k4_seq_4(k1_seq_4(A)) = A ) ) ), file(seq_4,t22_seq_4), [interesting(0.00)]). fof(d9_weierstr,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => k9_weierstr(A,B,C) = k5_weierstr(k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k8_weierstr(A,B),C)) ) ) ) ), file(weierstr,d9_weierstr), [interesting(0.00)]). fof(d6_weierstr,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr)) & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr)) ) => ( C = k6_weierstr(A,B) <=> ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => k1_funct_1(C,D) = k4_metric_1(A,D,B) ) ) ) ) ) ), file(weierstr,d6_weierstr), [interesting(0.00)]). fof(t5_metric_1,theorem,( ! [A] : ( ( v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => r1_xreal_0(0,k4_metric_1(A,B,C)) ) ) ) ), file(metric_1,t5_metric_1), [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(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(d8_weierstr,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr)) & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k3_topmetr)) ) => ( C = k8_weierstr(A,B) <=> ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => k1_funct_1(C,D) = k5_weierstr(k4_pre_topc(k5_pcomps_1(A),k3_topmetr,k6_weierstr(A,D),B)) ) ) ) ) ) ), file(weierstr,d8_weierstr), [interesting(0.00)]). fof(t143_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ( ~ r1_xreal_0(A,0) & r1_xreal_0(A,B) ) | ( r1_xreal_0(B,A) & ~ r1_xreal_0(0,A) ) ) => ( r1_xreal_0(k7_xcmplx_0(A,B),1) & r1_xreal_0(1,k7_xcmplx_0(B,A)) ) ) ) ) ), file(real_2,t143_real_2), [interesting(0.00)]). fof(t68_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(0,A) & r1_xreal_0(A,B) & r1_xreal_0(0,C) & r1_xreal_0(C,D) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t68_xreal_1), [interesting(0.00)]). fof(t36_euclid,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] : ( v1_xreal_0(D) => k18_euclid(D,A,k17_euclid(A,B,C)) = k17_euclid(A,k18_euclid(D,A,B),k18_euclid(D,A,C)) ) ) ) ) ), file(euclid,t36_euclid), [interesting(0.00)]). fof(t6_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))) => ( r2_hidden(B,k1_topreal1(A,B,C)) & r2_hidden(C,k1_topreal1(A,B,C)) ) ) ) ) ), file(topreal1,t6_topreal1), [interesting(0.00)]). fof(t121_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ( r1_xreal_0(0,A) & r1_xreal_0(0,B) ) | ( r1_xreal_0(A,0) & r1_xreal_0(B,0) ) ) => r1_xreal_0(0,k3_xcmplx_0(A,B)) ) ) ) ), file(real_2,t121_real_2), [interesting(0.00)]). fof(t31_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k17_euclid(A,k16_euclid(A),B) = B & k17_euclid(A,B,k16_euclid(A)) = B ) ) ) ), file(euclid,t31_euclid), [interesting(0.00)]). fof(t75_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k3_xcmplx_0(A,k7_xcmplx_0(B,C)) = k7_xcmplx_0(k3_xcmplx_0(A,B),C) ) ) ) ), file(xcmplx_1,t75_xcmplx_1), [interesting(0.00)]). fof(t50_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => r1_xreal_0(0,k6_xcmplx_0(B,A)) ) ) ) ), file(xreal_1,t50_xreal_1), [interesting(0.00)]). fof(t35_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(0,A) & r1_xreal_0(0,B) ) => r1_xreal_0(0,k2_xcmplx_0(A,B)) ) ) ) ), file(xreal_1,t35_xreal_1), [interesting(0.00)]). fof(t9_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), [interesting(0.00)]). fof(t37_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k18_euclid(k2_xcmplx_0(C,D),A,B) = k17_euclid(A,k18_euclid(C,A,B),k18_euclid(D,A,B)) ) ) ) ) ), file(euclid,t37_euclid), [interesting(0.00)]). fof(t50_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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))) => ~ ( C = k4_xboole_0(k1_euclid(A),a_2_3_jordan2c(A,B)) & r2_hidden(D,C) & r2_hidden(E,C) & ! [F] : ( m1_subset_1(F,k1_numbers) => ( D != k18_euclid(F,A,E) & E != k18_euclid(F,A,D) ) ) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(F,C) & r2_hidden(G,C) & r1_tarski(k3_topreal1(A,D,F),C) & r1_tarski(k3_topreal1(A,F,G),C) & r1_tarski(k3_topreal1(A,G,E),C) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t47_jordan2c,t48_jordan2c,d3_tarski,d4_xboole_0,t25_euclid,d10_xboole_0,d3_tarski,t25_euclid,d4_xboole_0,d3_tarski,d3_topreal1,t13_topreal3,d3_topreal1,d3_tarski,d12_funct_1,t13_topreal3,d6_weierstr,d12_funct_1,t13_topreal3,d6_weierstr,t5_metric_1,d2_seq_4,d1_funct_2,d6_weierstr,d12_funct_1,d5_seq_4,d3_weierstr,d5_weierstr,d8_weierstr,t45_jgraph_1,t13_jordan2c,t132_complex1,t26_toprns_1,t153_complex1,d1_absvalue,t143_real_2,t68_xreal_1,d9_xcmplx_0,d9_xcmplx_0,t11_absvalue,t2_xreal_1,t8_toprns_1,t36_euclid,t34_euclid,t34_euclid,t34_euclid,t34_euclid,t6_topreal1,d3_tarski,d3_topreal1,t127_real_2,t121_real_2,t33_euclid,t31_euclid,t33_euclid,t13_topreal3,d3_topreal1,d3_tarski,d12_funct_1,t13_topreal3,d6_weierstr,d12_funct_1,t13_topreal3,d6_weierstr,t5_metric_1,d2_seq_4,d1_funct_2,d6_weierstr,d12_funct_1,d5_seq_4,d3_weierstr,d5_weierstr,d8_weierstr,t45_jgraph_1,t13_jordan2c,t75_xcmplx_1,t88_xcmplx_1,t66_xreal_1,t50_xreal_1,t66_xreal_1,l5_jordan2c,d1_absvalue,t9_xreal_1,t8_toprns_1,t37_euclid,t34_euclid,t26_toprns_1,d3_tarski,d3_topreal1,t127_real_2,t121_real_2,t33_euclid,t31_euclid,t33_euclid,t13_topreal3,d3_topreal1,d3_tarski,d12_funct_1,t13_topreal3,d6_weierstr,d12_funct_1,t13_topreal3,d6_weierstr,t5_metric_1,d2_seq_4,d1_funct_2,d6_weierstr,d12_funct_1,d5_seq_4,d3_weierstr,d5_weierstr,d8_weierstr,t45_jgraph_1,t13_jordan2c,t75_xcmplx_1,t88_xcmplx_1,t66_xreal_1,t50_xreal_1,t66_xreal_1,l5_jordan2c,d1_absvalue,t9_xreal_1,t8_toprns_1,t37_euclid,t34_euclid,t26_toprns_1]), [file(jordan2c,t50_jordan2c),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(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(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(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(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(t4_funct_2,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( r1_tarski(k2_relat_1(B),A) => ( v1_funct_1(B) & v1_funct_2(B,k1_relat_1(B),A) & m2_relset_1(B,k1_relat_1(B),A) ) ) ) ), file(funct_2,t4_funct_2), [interesting(0.00)]). fof(t7_topmetr,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( ( v2_pre_topc(C) & l1_pre_topc(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) ) => ( ( v5_pre_topc(D,A,C) & m1_pre_topc(C,B) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) ) => ( E = D => v5_pre_topc(E,A,B) ) ) ) ) ) ) ) ), file(topmetr,t7_topmetr), [interesting(0.00)]). fof(d1_borsuk_2,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r1_borsuk_2(A,B,C) <=> ? [D] : ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) & v5_pre_topc(D,k5_topmetr,A) & k1_funct_1(D,0) = B & k1_funct_1(D,1) = C ) ) ) ) ) ), file(borsuk_2,d1_borsuk_2), [interesting(0.00)]). fof(d2_borsuk_2,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ( r1_borsuk_2(A,B,C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(A)) ) => ( m1_borsuk_2(D,A,B,C) <=> ( v5_pre_topc(D,k5_topmetr,A) & k1_funct_1(D,0) = B & k1_funct_1(D,1) = C ) ) ) ) ) ) ) ), file(borsuk_2,d2_borsuk_2), [interesting(0.00)]). fof(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(t46_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => ( r1_tarski(k2_relat_1(A),k1_relat_1(B)) => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ), file(relat_1,t46_relat_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(t2_treal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) & r1_xreal_0(B,C) ) => k4_subset_1(k1_numbers,k1_rcomp_1(A,C),k1_rcomp_1(B,D)) = k1_rcomp_1(A,D) ) ) ) ) ) ), file(treal_1,t2_treal_1), [interesting(0.00)]). fof(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(d17_borsuk_1,definition,( k23_borsuk_1 = 0 ), file(borsuk_1,d17_borsuk_1), [interesting(0.00)]). fof(d18_borsuk_1,definition,( k24_borsuk_1 = 1 ), file(borsuk_1,d18_borsuk_1), [interesting(0.00)]). fof(t8_treal_1,theorem, ( k23_borsuk_1 = k1_treal_1(0,1) & k24_borsuk_1 = k2_treal_1(0,1) ), file(treal_1,t8_treal_1), [interesting(0.00)]). fof(t14_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))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k4_topmetr(A,B))) => ! [F] : ( m1_subset_1(F,k1_numbers) => ! [G] : ( m1_subset_1(G,k1_numbers) => ! [H] : ( m1_subset_1(H,k1_numbers) => ( ( E = F & G = C & H = D ) => k8_funct_2(u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(0,1)),k4_treal_1(A,B,C,D),E) = k2_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(k5_real_1(H,G),k6_xcmplx_0(B,A)),F),k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(B,G),k3_xcmplx_0(A,H)),k6_xcmplx_0(B,A))) ) ) ) ) ) ) ) ) ) ) ), file(treal_1,t14_treal_1), [interesting(0.00)]). fof(t23_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ), file(funct_1,t23_funct_1), [interesting(0.00)]). fof(t12_funct_4,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( ~ r2_hidden(A,k1_relat_1(B)) => k1_funct_1(k1_funct_4(C,B),A) = k1_funct_1(C,A) ) ) ) ), file(funct_4,t12_funct_4), [interesting(0.00)]). fof(t45_relat_1,theorem,( ! [A] : ( v1_relat_1(A) => ! [B] : ( v1_relat_1(B) => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ), file(relat_1,t45_relat_1), [interesting(0.00)]). fof(t18_funct_4,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => r1_tarski(k2_relat_1(k1_funct_4(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) ), file(funct_4,t18_funct_4), [interesting(0.00)]). fof(t13_xboole_1,theorem,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) ) => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,D)) ) ), file(xboole_1,t13_xboole_1), [interesting(0.00)]). fof(t11_relset_1,theorem,( ! [A,B,C] : ( v1_relat_1(C) => ( ( r1_tarski(k1_relat_1(C),A) & r1_tarski(k2_relat_1(C),B) ) => m2_relset_1(C,A,B) ) ) ), file(relset_1,t11_relset_1), [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(t15_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))) => ( v1_funct_1(k4_treal_1(A,B,C,D)) & v1_funct_2(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(0,1))) & v5_pre_topc(k4_treal_1(A,B,C,D),k4_topmetr(A,B),k4_topmetr(0,1)) & m2_relset_1(k4_treal_1(A,B,C,D),u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k4_topmetr(0,1))) ) ) ) ) ) ) ), file(treal_1,t15_treal_1), [interesting(0.00)]). fof(t19_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,B,C) & m2_relset_1(E,B,C) ) => ( ~ ( B = k1_xboole_0 & C != k1_xboole_0 & A != k1_xboole_0 ) => ( v1_funct_1(k1_partfun1(A,B,B,C,D,E)) & v1_funct_2(k1_partfun1(A,B,B,C,D,E),A,C) & m2_relset_1(k1_partfun1(A,B,B,C,D,E),A,C) ) ) ) ) ), file(funct_2,t19_funct_2), [interesting(0.00)]). fof(t58_tops_2,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ! [C] : ( ( ~ v3_struct_0(C) & l1_pre_topc(C) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(C)) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(C)) ) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(C),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(C),u1_struct_0(B)) ) => ( ( v5_pre_topc(D,A,C) & v5_pre_topc(E,C,B) ) => v5_pre_topc(k7_funct_2(u1_struct_0(A),u1_struct_0(C),u1_struct_0(B),D,E),A,B) ) ) ) ) ) ) ), file(tops_2,t58_tops_2), [interesting(0.00)]). fof(t1_topmetr2,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) ) => k5_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_rcomp_1(B,C)) = k1_tarski(B) ) ) ) ) ), file(topmetr2,t1_topmetr2), [interesting(0.00)]). fof(t38_pcomps_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => v3_compts_1(k5_pcomps_1(A)) ) ), file(pcomps_1,t38_pcomps_1), [interesting(0.00)]). fof(d7_topmetr,definition,( k3_topmetr = k5_pcomps_1(k8_metric_1) ), file(topmetr,d7_topmetr), [interesting(0.00)]). fof(t11_heine,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => v2_compts_1(k4_topmetr(A,B)) ) ) ) ), file(heine,t11_heine), [interesting(0.00)]). fof(t3_topmetr,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & m1_pre_topc(B,A) ) => ( v3_compts_1(A) => v3_compts_1(B) ) ) ) ), file(topmetr,t3_topmetr), [interesting(0.00)]). fof(t4_topmetr2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v2_pre_topc(B) & l1_pre_topc(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_pre_topc(D,A) => ! [E] : ( m1_pre_topc(E,A) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(D),u1_struct_0(B)) & m2_relset_1(F,u1_struct_0(D),u1_struct_0(B)) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(E),u1_struct_0(B)) & m2_relset_1(G,u1_struct_0(E),u1_struct_0(B)) ) => ( ( k2_xboole_0(k2_pre_topc(D),k2_pre_topc(E)) = k2_pre_topc(A) & k3_xboole_0(k2_pre_topc(D),k2_pre_topc(E)) = k1_struct_0(A,C) & v2_compts_1(D) & v2_compts_1(E) & v3_compts_1(A) & v5_pre_topc(F,D,B) & v5_pre_topc(G,E,B) & k1_funct_1(F,C) = k1_funct_1(G,C) ) => ( v1_funct_1(k1_funct_4(F,G)) & v1_funct_2(k1_funct_4(F,G),u1_struct_0(A),u1_struct_0(B)) & v5_pre_topc(k1_funct_4(F,G),A,B) & m2_relset_1(k1_funct_4(F,G),u1_struct_0(A),u1_struct_0(B)) ) ) ) ) ) ) ) ) ) ), file(topmetr2,t4_topmetr2), [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(d5_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), [interesting(0.00)]). fof(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(t21_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(k2_xcmplx_0(A,B),C) <=> r1_xreal_0(A,k6_xcmplx_0(C,B)) ) ) ) ) ), file(xreal_1,t21_xreal_1), [interesting(0.00)]). fof(t22_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,k2_xcmplx_0(B,C)) <=> r1_xreal_0(k6_xcmplx_0(A,B),C) ) ) ) ) ), file(xreal_1,t22_xreal_1), [interesting(0.00)]). fof(t4_borsuk_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ? [C] : ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(A)) & v5_pre_topc(C,k5_topmetr,A) & k1_funct_1(C,0) = B & k1_funct_1(C,1) = B ) ) ) ), file(borsuk_2,t4_borsuk_2), [interesting(0.00)]). fof(t24_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k5_toprns_1(A,k16_euclid(A)) = 0 ) ), file(toprns_1,t24_toprns_1), [interesting(0.00)]). fof(t2_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => k3_finseq_1(B) = A ) ) ), file(euclid,t2_euclid), [interesting(0.00)]). fof(d1_relat_1,definition,( ! [A] : ( v1_relat_1(A) <=> ! [B] : ~ ( r2_hidden(B,A) & ! [C,D] : B != k4_tarski(C,D) ) ) ), file(relat_1,d1_relat_1), [interesting(0.00)]). fof(d4_relat_1,definition,( ! [A] : ( v1_relat_1(A) => ! [B] : ( B = k1_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : r2_hidden(k4_tarski(C,D),A) ) ) ) ), file(relat_1,d4_relat_1), [interesting(0.00)]). fof(d3_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( B = k3_finseq_1(A) <=> k2_finseq_1(B) = k1_relat_1(A) ) ) ) ), file(finseq_1,d3_finseq_1), [interesting(0.00)]). fof(t3_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_1), [interesting(0.00)]). fof(t8_funct_1,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(k4_tarski(A,B),C) <=> ( r2_hidden(A,k1_relat_1(C)) & B = k1_funct_1(C,A) ) ) ) ), file(funct_1,t8_funct_1), [interesting(0.00)]). fof(d2_zfmisc_1,definition,( ! [A,B,C] : ( C = k2_zfmisc_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ? [E,F] : ( r2_hidden(E,A) & r2_hidden(F,B) & D = k4_tarski(E,F) ) ) ) ), file(zfmisc_1,d2_zfmisc_1), [interesting(0.00)]). fof(d2_funcop_1,definition,( ! [A,B] : k2_funcop_1(A,B) = k2_zfmisc_1(A,k1_tarski(B)) ), file(funcop_1,d2_funcop_1), [interesting(0.00)]). fof(d2_finseq_2,definition,( ! [A] : ( v4_ordinal2(A) => ! [B] : k2_finseq_2(A,B) = k2_funcop_1(k2_finseq_1(A),B) ) ), file(finseq_2,d2_finseq_2), [interesting(0.00)]). fof(t54_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ~ ( B != k5_euclid(A) & B = C & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(1,D) & r1_xreal_0(D,A) & k1_funct_1(C,D) != 0 ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_euclid,d1_relat_1,d4_relat_1,d3_finseq_1,t3_finseq_1,t8_funct_1,d1_tarski,d1_tarski,t3_finseq_1,d3_finseq_1,t8_funct_1,d2_zfmisc_1,d2_funcop_1,d2_finseq_2]), [file(jordan2c,t54_jordan2c),interesting(0.00)]). fof(t35_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k3_finseq_1(k7_finseq_1(A,B)) = k1_nat_1(k3_finseq_1(A),k3_finseq_1(B)) ) ) ), file(finseq_1,t35_finseq_1), [interesting(0.00)]). fof(t56_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k2_relat_1(B) = k1_tarski(A) ) ) ) ), file(finseq_1,t56_finseq_1), [interesting(0.00)]). fof(t27_jordan3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(B)) & r1_xreal_0(1,D) & r1_xreal_0(D,k3_finseq_1(B)) ) => ( k1_funct_1(k1_jordan3(A,B,C,D),1) = k1_funct_1(B,C) & ( r1_xreal_0(C,D) => ( k3_finseq_1(k1_jordan3(A,B,C,D)) = k1_nat_1(k5_binarith(D,C),1) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,E) & r1_xreal_0(E,k3_finseq_1(k1_jordan3(A,B,C,D))) ) => k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k5_binarith(k1_nat_1(E,C),1)) ) ) ) ) & ( ~ r1_xreal_0(C,D) => ( k3_finseq_1(k1_jordan3(A,B,C,D)) = k1_nat_1(k5_binarith(C,D),1) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,E) & r1_xreal_0(E,k3_finseq_1(k1_jordan3(A,B,C,D))) ) => k1_funct_1(k1_jordan3(A,B,C,D),E) = k1_funct_1(B,k1_nat_1(k5_binarith(C,E),1)) ) ) ) ) ) ) ) ) ) ) ), file(jordan3,t27_jordan3), [interesting(0.00)]). fof(t50_binarith,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(A,B) => k5_binarith(B,A) = k6_xcmplx_0(B,A) ) ) ) ), file(binarith,t50_binarith), [interesting(0.00)]). fof(d11_finseq_1,definition,( ! [A,B] : ( B = k13_finseq_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> m2_finseq_1(C,A) ) ) ), file(finseq_1,d11_finseq_1), [interesting(0.00)]). fof(d4_finseq_2,definition,( ! [A] : ( v4_ordinal2(A) => ! [B] : k4_finseq_2(A,B) = a_2_0_finseq_2(A,B) ) ), file(finseq_2,d4_finseq_2), [interesting(0.00)]). fof(t24_finseq_4,theorem,( ! [A,B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,k3_finseq_1(B)) ) => k4_finseq_4(k5_numbers,A,B,C) = k1_funct_1(B,C) ) ) ) ), file(finseq_4,t24_finseq_4), [interesting(0.00)]). fof(t12_jordan3,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ ( r1_xreal_0(B,A) & ~ r1_xreal_0(k2_xcmplx_0(A,1),B) ) => r1_xreal_0(A,k5_binarith(B,1)) ) ) ) ), file(jordan3,t12_jordan3), [interesting(0.00)]). fof(t11_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), [interesting(0.00)]). fof(t39_binarith,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => k5_binarith(k2_xcmplx_0(A,B),B) = A ) ) ), file(binarith,t39_binarith), [interesting(0.00)]). fof(t36_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ! [C] : ( v4_ordinal2(C) => ( ( r1_xreal_0(k1_nat_1(k3_finseq_1(A),1),C) & r1_xreal_0(C,k1_nat_1(k3_finseq_1(A),k3_finseq_1(B))) ) => k1_funct_1(k7_finseq_1(A,B),C) = k1_funct_1(B,k6_xcmplx_0(C,k3_finseq_1(A))) ) ) ) ) ), file(finseq_1,t36_finseq_1), [interesting(0.00)]). fof(d7_rvsum_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_finseq_1(B,k1_numbers) => k9_rvsum_1(A,B) = k5_finseqop(k1_numbers,k1_numbers,B,k2_rvsum_1(A)) ) ) ), file(rvsum_1,d7_rvsum_1), [interesting(0.00)]). fof(d3_rvsum_1,definition,( ! [A] : ( v1_xreal_0(A) => k2_rvsum_1(A) = k5_funcop_1(k35_binop_2,A,k6_partfun1(k1_numbers)) ) ), file(rvsum_1,d3_rvsum_1), [interesting(0.00)]). fof(d5_funcop_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => k5_funcop_1(A,B,C) = k5_relat_1(k13_funct_3(k2_funcop_1(k1_relat_1(C),B),C),A) ) ) ), file(funcop_1,d5_funcop_1), [interesting(0.00)]). fof(d8_funct_3,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 = k13_funct_3(A,B) <=> ( k1_relat_1(C) = k3_xboole_0(k1_relat_1(A),k1_relat_1(B)) & ! [D] : ( r2_hidden(D,k1_relat_1(C)) => k1_funct_1(C,D) = k4_tarski(k1_funct_1(A,D),k1_funct_1(B,D)) ) ) ) ) ) ) ), file(funct_3,d8_funct_3), [interesting(0.00)]). fof(t22_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B))) => k1_funct_1(k5_relat_1(C,B),A) = k1_funct_1(B,k1_funct_1(C,A)) ) ) ) ), file(funct_1,t22_funct_1), [interesting(0.00)]). fof(t34_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ( B = k6_relat_1(A) <=> ( k1_relat_1(B) = A & ! [C] : ( r2_hidden(C,A) => k1_funct_1(B,C) = C ) ) ) ) ), file(funct_1,t34_funct_1), [interesting(0.00)]). fof(t19_funcop_1,theorem,( ! [A,B] : ( k1_relat_1(k2_funcop_1(A,B)) = A & r1_tarski(k2_relat_1(k2_funcop_1(A,B)),k1_tarski(B)) ) ), file(funcop_1,t19_funcop_1), [interesting(0.00)]). fof(t13_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(B,A) => k1_funct_1(k2_funcop_1(A,C),B) = C ) ), file(funcop_1,t13_funcop_1), [interesting(0.00)]). fof(d11_binop_2,definition,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ) => ( A = k35_binop_2 <=> ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => k2_binop_1(k1_numbers,k1_numbers,k1_numbers,A,B,C) = k11_binop_2(B,C) ) ) ) ) ), file(binop_2,d11_binop_2), [interesting(0.00)]). fof(t53_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ! [C] : ( m2_finseq_1(C,k1_numbers) => ! [D] : ( m2_finseq_1(D,k1_numbers) => ! [E] : ( m1_subset_1(E,k1_numbers) => ( ( C = B & D = k9_euclid(A,E,B) ) => ( k3_finseq_1(C) = k3_finseq_1(D) & ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,F) & r1_xreal_0(F,k3_finseq_1(C)) ) => k4_finseq_4(k5_numbers,k1_numbers,D,F) = k11_binop_2(E,k4_finseq_4(k5_numbers,k1_numbers,C,F)) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_euclid,t2_euclid,d7_rvsum_1,d3_rvsum_1,d5_funcop_1,d8_funct_3,t3_finseq_1,d3_finseq_1,t22_funct_1,t34_funct_1,t34_funct_1,t24_finseq_4,t19_funcop_1,t34_funct_1,d8_funct_3,t13_funcop_1,t34_funct_1,t23_funct_1,d11_binop_2,t24_finseq_4,t2_euclid]), [file(jordan2c,t53_jordan2c),interesting(0.00)]). fof(t5_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => ( k3_xcmplx_0(B,A) = k3_xcmplx_0(C,A) => ( A = 0 | B = C ) ) ) ) ) ), file(xcmplx_1,t5_xcmplx_1), [interesting(0.00)]). fof(t58_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k1_funct_1(k7_finseq_1(k9_finseq_1(A),B),1) = A ) ), file(finseq_1,t58_finseq_1), [interesting(0.00)]). fof(t7_jordan3,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r1_xreal_0(A,B) => r1_xreal_0(k5_binarith(A,1),B) ) ) ) ), file(jordan3,t7_jordan3), [interesting(0.00)]). fof(t17_jordan3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B,C] : ( m2_finseq_1(C,A) => ( r1_xreal_0(1,k3_finseq_1(C)) => ( k1_funct_1(k7_finseq_1(C,k9_finseq_1(B)),1) = k1_funct_1(C,1) & k1_funct_1(k7_finseq_1(C,k9_finseq_1(B)),1) = k4_finseq_4(k5_numbers,A,C,1) & k1_funct_1(k7_finseq_1(k9_finseq_1(B),C),k1_nat_1(k3_finseq_1(C),1)) = k1_funct_1(C,k3_finseq_1(C)) & k1_funct_1(k7_finseq_1(k9_finseq_1(B),C),k1_nat_1(k3_finseq_1(C),1)) = k4_finseq_4(k5_numbers,A,C,k3_finseq_1(C)) ) ) ) ) ), file(jordan3,t17_jordan3), [interesting(0.00)]). fof(t32_jordan3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,C) & r1_xreal_0(C,k3_finseq_1(B)) ) => k1_funct_1(k1_jordan3(A,B,1,C),D) = k1_funct_1(B,D) ) ) ) ) ) ), file(jordan3,t32_jordan3), [interesting(0.00)]). fof(t59_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => k1_funct_1(k7_finseq_1(B,k9_finseq_1(A)),k1_nat_1(k3_finseq_1(B),1)) = A ) ), file(finseq_1,t59_finseq_1), [interesting(0.00)]). fof(t55_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ~ ( r1_xreal_0(2,A) & B != k5_euclid(A) & ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => ~ ! [D] : ( m1_subset_1(D,k1_numbers) => ( C != k9_euclid(A,D,B) & B != k9_euclid(A,D,C) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t54_jordan2c,t2_euclid,t2_xreal_1,t35_finseq_1,t56_finseq_1,t27_jordan3,t50_binarith,t56_finseq_1,t52_jordan2c,t24_finseq_4,t24_finseq_4,t2_xreal_1,t50_binarith,t12_jordan3,t2_euclid,t11_xreal_1,t39_binarith,t50_binarith,t2_euclid,t38_nat_1,t36_finseq_1,t27_jordan3,t53_jordan2c,t5_xcmplx_1,t53_jordan2c,t24_finseq_4,t58_finseq_1,t24_finseq_4,t24_finseq_4,t2_xreal_1,t50_binarith,t12_jordan3,t2_euclid,t11_xreal_1,t39_binarith,t50_binarith,t2_euclid,t38_nat_1,t36_finseq_1,t27_jordan3,t53_jordan2c,t5_xcmplx_1,t53_jordan2c,t24_finseq_4,t58_finseq_1,t1_xreal_1,t56_finseq_1,t50_binarith,t7_jordan3,t11_xreal_1,t50_binarith,t27_jordan3,t27_jordan3,t35_finseq_1,t52_jordan2c,t24_finseq_4,t24_finseq_4,t2_xreal_1,t17_jordan3,t32_jordan3,t53_jordan2c,t5_xcmplx_1,t53_jordan2c,t24_finseq_4,t59_finseq_1,t24_finseq_4,t24_finseq_4,t2_xreal_1,t17_jordan3,t32_jordan3,t53_jordan2c,t5_xcmplx_1,t53_jordan2c,t24_finseq_4,t59_finseq_1]), [file(jordan2c,t55_jordan2c),interesting(0.00)]). fof(t12_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => r1_xreal_0(0,k12_euclid(B)) ) ) ), file(euclid,t12_euclid), [interesting(0.00)]). fof(t11_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k1_euclid(A)) => ( k12_euclid(B) = 0 => B = k5_euclid(A) ) ) ) ), file(euclid,t11_euclid), [interesting(0.00)]). fof(d11_euclid,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(B))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(B))) => ( D = k18_euclid(A,B,C) <=> ! [E] : ( m2_finseq_2(E,k1_numbers,k1_euclid(B)) => ( E = C => D = k9_euclid(B,A,E) ) ) ) ) ) ) ) ), file(euclid,d11_euclid), [interesting(0.00)]). fof(t14_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( m2_finseq_2(C,k1_numbers,k1_euclid(A)) => k12_euclid(k9_euclid(A,B,C)) = k4_real_1(k18_complex1(B),k12_euclid(C)) ) ) ) ), file(euclid,t14_euclid), [interesting(0.00)]). fof(t71_rvsum_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( m2_finseq_2(D,k1_numbers,k4_finseq_2(A,k1_numbers)) => k10_rvsum_1(A,k11_binop_2(B,C),D) = k10_rvsum_1(A,B,k10_rvsum_1(A,C,D)) ) ) ) ) ), file(rvsum_1,t71_rvsum_1), [interesting(0.00)]). fof(t74_rvsum_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_2(B,k1_numbers,k4_finseq_2(A,k1_numbers)) => k10_rvsum_1(A,1,B) = B ) ) ), file(rvsum_1,t74_rvsum_1), [interesting(0.00)]). fof(t7_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( r2_hidden(C,A) => ( B = k1_xboole_0 | r2_hidden(k1_funct_1(D,C),B) ) ) ) ), file(funct_2,t7_funct_2), [interesting(0.00)]). fof(t43_jordan2c,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] : ( ( ~ v1_xboole_0(E) & m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) ) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) & m2_relset_1(G,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) ) => ~ ( v5_pre_topc(F,k5_topmetr,k3_pre_topc(k15_euclid(A),E)) & B = k1_funct_1(F,0) & C = k1_funct_1(F,1) & v5_pre_topc(G,k5_topmetr,k3_pre_topc(k15_euclid(A),E)) & C = k1_funct_1(G,0) & D = k1_funct_1(G,1) & ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) & m2_relset_1(H,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) ) => ~ ( v5_pre_topc(H,k5_topmetr,k3_pre_topc(k15_euclid(A),E)) & B = k1_funct_1(H,0) & D = k1_funct_1(H,1) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t48_rcomp_1,t48_rcomp_1,t83_borsuk_1,t7_funct_2,d1_borsuk_2,d2_borsuk_2,d1_borsuk_2,d2_borsuk_2,d1_funct_2,t25_topmetr,d1_funct_2,t25_topmetr,d1_funct_2,t27_topmetr,t46_relat_1,t27_topmetr,d1_funct_4,t46_relat_1,t83_borsuk_1,t2_treal_1,d1_funct_2,t25_topmetr,d1_rcomp_1,d1_funct_2,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t14_treal_1,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t23_funct_1,d1_rcomp_1,t12_funct_4,t45_relat_1,t1_xboole_1,t18_funct_4,t13_xboole_1,t1_xboole_1,d1_funct_2,t11_relset_1,t27_topmetr,t6_treal_1,t15_treal_1,t15_treal_1,t19_funct_2,t27_topmetr,t19_funct_2,t27_topmetr,t19_funct_2,t27_topmetr,t83_borsuk_1,d1_rcomp_1,t27_topmetr,t58_tops_2,t12_pre_topc,t25_topmetr,t12_pre_topc,t25_topmetr,t2_treal_1,t83_borsuk_1,t12_pre_topc,d1_funct_2,t25_topmetr,d1_rcomp_1,d1_funct_2,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t14_treal_1,d17_borsuk_1,d18_borsuk_1,t8_treal_1,t23_funct_1,t1_topmetr2,t38_pcomps_1,d7_topmetr,t11_heine,t3_topmetr,t4_topmetr2,d1_rcomp_1,t14_funct_4,d1_borsuk_2,d2_borsuk_2,d1_rcomp_1,d3_xboole_0,t1_topmetr2,d1_tarski,t12_funct_4,d1_rcomp_1,t25_topmetr,d1_funct_2,t14_funct_4,d1_rcomp_1,t14_funct_4]), [file(jordan2c,t43_jordan2c),interesting(0.00)]). fof(t5_jordan1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => ~ ( r2_hidden(C,B) & r2_hidden(D,B) & C != D & ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B))) & m2_relset_1(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(A,B))) ) => ~ ( v5_pre_topc(E,k22_borsuk_1,k3_pre_topc(A,B)) & C = k1_funct_1(E,0) & D = k1_funct_1(E,1) ) ) ) ) ) => v2_connsp_1(B,A) ) ) ) ), file(jordan1,t5_jordan1), [interesting(0.00)]). fof(t8_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(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), [interesting(0.00)]). fof(t4_metric_1,theorem,( ! [A] : ( l1_metric_1(A) => ( ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => r1_xreal_0(k2_metric_1(A,B,D),k3_real_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) ) <=> v9_metric_1(A) ) ) ), file(metric_1,t4_metric_1), [interesting(0.00)]). fof(t22_subset_1,theorem,( ! [A] : k2_subset_1(A) = k3_subset_1(A,k1_subset_1(A)) ), file(subset_1,t22_subset_1), [interesting(0.00)]). fof(d4_subset_1,definition,( ! [A] : k2_subset_1(A) = A ), file(subset_1,d4_subset_1), [interesting(0.00)]). fof(t32_subset_1,theorem,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => k6_subset_1(A,B,C) = k5_subset_1(A,B,k3_subset_1(A,C)) ) ) ), file(subset_1,t32_subset_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(t43_subset_1,theorem,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => ( r1_xboole_0(B,C) <=> r1_tarski(B,k3_subset_1(A,C)) ) ) ) ), file(subset_1,t43_subset_1), [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(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(t9_connsp_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_connsp_1(B,A) => ( B = k1_xboole_0 | r3_connsp_1(A,k1_connsp_3(A,B)) ) ) ) ) ), file(connsp_3,t9_connsp_3), [interesting(0.00)]). fof(t1_connsp_3,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))) & v2_connsp_1(C,A) & r1_tarski(B,C) ) => r1_tarski(B,k1_connsp_3(A,B)) ) ) ) ), file(connsp_3,t1_connsp_3), [interesting(0.00)]). fof(t21_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(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_tbsp_1(B,A) & r1_tarski(C,B) ) => v6_tbsp_1(C,A) ) ) ) ) ), file(tbsp_1,t21_tbsp_1), [interesting(0.00)]). fof(d4_jordan2c,definition,( ! [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)))) => ( r2_jordan2c(A,B,C) <=> ( r4_connsp_1(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B),C) & ~ v1_jordan2c(C,A) ) ) ) ) ) ), file(jordan2c,d4_jordan2c), [interesting(0.00)]). fof(t18_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)))) => ( r2_jordan2c(A,B,C) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B))))) & D = C & r3_connsp_1(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B)),D) & ~ ( v6_tbsp_1(D,k14_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d6_connsp_1,d4_jordan2c,d4_jordan2c,t13_topreal3,d2_jordan2c,d2_jordan2c,d6_connsp_1,d4_jordan2c]), [file(jordan2c,t18_jordan2c),interesting(0.00)]). fof(t36_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(0,A) & ~ r1_xreal_0(B,0) & r1_xreal_0(k2_xcmplx_0(A,B),0) ) ) ) ), file(xreal_1,t36_xreal_1), [interesting(0.00)]). fof(t40_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)))) => ( v1_jordan2c(B,A) <=> ? [C] : ( m1_subset_1(C,k1_numbers) & ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(D,B) & r1_xreal_0(C,k5_toprns_1(A,D)) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d2_jordan2c,d9_tbsp_1,t13_topreal3,t45_jgraph_1,t13_jordan2c,t4_metric_1,t8_xreal_1,t2_xreal_1,t31_xreal_1,t2_xreal_1,t13_topreal3,t13_topreal3,t26_toprns_1,l6_jordan2c,t4_metric_1,t45_jgraph_1,t13_jordan2c,t45_jgraph_1,t13_jordan2c,t9_xreal_1,t2_xreal_1,d9_tbsp_1,d2_jordan2c]), [file(jordan2c,t40_jordan2c),interesting(0.00)]). fof(t75_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ( ? [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) & C = a_2_3_jordan2c(A,B) ) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ( C = a_2_3_jordan2c(A,B) => v6_tbsp_1(C,k14_euclid(A)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t13_topreal3,t13_topreal3,t40_jordan2c,d2_jordan2c]), [file(jordan2c,t75_jordan2c),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(d5_connsp_1,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r3_connsp_1(A,B) <=> ( v2_connsp_1(B,A) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v2_connsp_1(C,A) & r1_tarski(B,C) ) => B = C ) ) ) ) ) ) ), file(connsp_1,d5_connsp_1), [interesting(0.00)]). fof(t30_toprns_1,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))) => r1_xreal_0(k5_toprns_1(A,k17_euclid(A,B,C)),k3_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C))) ) ) ) ), file(toprns_1,t30_toprns_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(t74_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)))) => ~ ( r1_xreal_0(1,A) & v1_jordan2c(B,A) & k3_subset_1(u1_struct_0(k15_euclid(A)),B) = k1_xboole_0 ) ) ) ), inference(mizar_proof,[status(thm)],[t40_jordan2c,t8_toprns_1,t38_jordan2c,t132_complex1,t66_xreal_1,t11_absvalue,t2_xreal_1,d4_xboole_0,d5_subset_1]), [file(jordan2c,t74_jordan2c),interesting(0.00)]). fof(t16_connsp_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v2_connsp_1(B,A) & v2_connsp_1(C,A) ) => ( r1_xboole_0(B,C) | ( r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k1_connsp_3(A,B)) & r1_tarski(k4_subset_1(u1_struct_0(A),B,C),k1_connsp_3(A,C)) & r1_tarski(B,k1_connsp_3(A,C)) & r1_tarski(C,k1_connsp_3(A,B)) ) ) ) ) ) ) ), file(connsp_3,t16_connsp_3), [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(t27_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => k2_pre_topc(A) = k3_subset_1(u1_struct_0(A),k1_pre_topc(A)) ) ), file(pre_topc,t27_pre_topc), [interesting(0.00)]). fof(t3_tsep_1,theorem,( ! [A] : ( ( v1_pre_topc(A) & l1_pre_topc(A) ) => k3_pre_topc(A,k2_pre_topc(A)) = A ) ), file(tsep_1,t3_tsep_1), [interesting(0.00)]). fof(t28_connsp_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r1_tarski(B,C) => k4_connsp_3(A,B,C) = B ) ) ) ) ), file(connsp_3,t28_connsp_3), [interesting(0.00)]). fof(t7_connsp_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r3_connsp_1(A,B) => k1_connsp_3(A,B) = B ) ) ) ), file(connsp_3,t7_connsp_3), [interesting(0.00)]). fof(t11_gobrd12,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => k3_subset_1(u1_struct_0(k15_euclid(2)),k5_topreal1(2,A)) = k4_subset_1(u1_struct_0(k15_euclid(2)),k2_goboard9(A),k3_goboard9(A)) ) ), file(gobrd12,t11_gobrd12), [interesting(0.00)]). fof(d2_goboard9,definition,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( B = k3_goboard9(A) <=> ( r4_connsp_1(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),k5_topreal1(2,A)),B) & r1_tarski(k1_tops_1(k15_euclid(2),k4_goboard5(A,1)),B) ) ) ) ) ), file(goboard9,d2_goboard9), [interesting(0.00)]). fof(t96_sprect_1,theorem,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => r1_xboole_0(k2_goboard9(A),k3_goboard9(A)) ) ), file(sprect_1,t96_sprect_1), [interesting(0.00)]). fof(d1_goboard9,definition,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( B = k2_goboard9(A) <=> ( r4_connsp_1(k15_euclid(2),k3_subset_1(u1_struct_0(k15_euclid(2)),k5_topreal1(2,A)),B) & r1_tarski(k1_tops_1(k15_euclid(2),k5_goboard5(A,1)),B) ) ) ) ) ), file(goboard9,d1_goboard9), [interesting(0.00)]). fof(t32_toprns_1,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))) => r1_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k17_euclid(A,B,C))) ) ) ) ), file(toprns_1,t32_toprns_1), [interesting(0.00)]). fof(t60_square_1,theorem,( k7_square_1(0) = 0 ), file(square_1,t60_square_1), [interesting(0.00)]). fof(t89_square_1,theorem,( ! [A] : ( v1_xreal_0(A) => ( r1_xreal_0(0,A) => k8_square_1(k5_square_1(A)) = A ) ) ), file(square_1,t89_square_1), [interesting(0.00)]). fof(t90_sprect_1,theorem,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => k3_finseq_1(A) = 5 ) ), file(sprect_1,t90_sprect_1), [interesting(0.00)]). fof(t33_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v2_sppol_1(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ r1_xreal_0(k20_pscomp_1(A),k18_pscomp_1(A)) ) ), file(sprect_1,t33_sprect_1), [interesting(0.00)]). fof(t34_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v1_sppol_1(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ~ r1_xreal_0(k19_pscomp_1(A),k21_pscomp_1(A)) ) ), file(sprect_1,t34_sprect_1), [interesting(0.00)]). fof(t91_sprect_1,theorem,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k31_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t91_sprect_1), [interesting(0.00)]). fof(t92_sprect_1,theorem,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,2) = k33_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,2) = k34_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t92_sprect_1), [interesting(0.00)]). fof(t93_sprect_1,theorem,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,3) = k36_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,3) = k35_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t93_sprect_1), [interesting(0.00)]). fof(t94_sprect_1,theorem,( ! [A] : ( ( v1_sprect_1(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,4) = k37_pscomp_1(k5_topreal1(2,A)) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,4) = k30_pscomp_1(k5_topreal1(2,A)) ) ) ), file(sprect_1,t94_sprect_1), [interesting(0.00)]). fof(d1_finseq_6,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_1(B,A) => ( v1_finseq_6(B,A) <=> k4_finseq_4(k5_numbers,A,B,1) = k4_finseq_4(k5_numbers,A,B,k3_finseq_1(B)) ) ) ) ), file(finseq_6,d1_finseq_6), [interesting(0.00)]). fof(t55_sprect_2,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & ~ v5_seqm_3(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ~ r1_xreal_0(k21_euclid(k33_pscomp_1(k5_topreal1(2,A))),k21_euclid(k32_pscomp_1(k5_topreal1(2,A)))) ) ), file(sprect_2,t55_sprect_2), [interesting(0.00)]). fof(d3_goboard2,definition,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => k3_goboard2(A) = k1_goboard2(k2_goboard2(k2_goboard1(A)),k2_goboard2(k3_goboard1(A))) ) ), file(goboard2,d3_goboard2), [interesting(0.00)]). fof(t19_sprect_2,theorem,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => k4_finseq_1(k2_goboard1(A)) = k4_finseq_1(A) ) ), file(sprect_2,t19_sprect_2), [interesting(0.00)]). fof(d3_goboard1,definition,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ( B = k2_goboard1(A) <=> ( k3_finseq_1(B) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(B)) => k1_goboard1(B,C) = k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C)) ) ) ) ) ) ) ), file(goboard1,d3_goboard1), [interesting(0.00)]). fof(t84_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k21_euclid(k30_pscomp_1(A)) = k18_pscomp_1(A) & k21_euclid(k31_pscomp_1(A)) = k18_pscomp_1(A) ) ) ), file(pscomp_1,t84_pscomp_1), [interesting(0.00)]). fof(t104_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k21_euclid(k35_pscomp_1(A)) = k20_pscomp_1(A) & k21_euclid(k34_pscomp_1(A)) = k20_pscomp_1(A) ) ) ), file(pscomp_1,t104_pscomp_1), [interesting(0.00)]). fof(t27_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(B,k2_xcmplx_0(A,1)) & A != B & B != k2_xcmplx_0(A,1) ) ) ) ), file(nat_1,t27_nat_1), [interesting(0.00)]). fof(t2_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(B,k23_binop_2(A,2)) & B != A & B != k23_binop_2(A,1) & B != k23_binop_2(A,2) ) ) ) ), inference(mizar_proof,[status(thm)],[t27_nat_1,t27_nat_1]), [file(jordan2c,t2_jordan2c),interesting(0.00)]). fof(t3_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(B,k23_binop_2(A,3)) & B != A & B != k23_binop_2(A,1) & B != k23_binop_2(A,2) & B != k23_binop_2(A,3) ) ) ) ), inference(mizar_proof,[status(thm)],[t2_jordan2c,t27_nat_1]), [file(jordan2c,t3_jordan2c),interesting(0.00)]). fof(t4_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,B) & r1_xreal_0(B,k23_binop_2(A,4)) & B != A & B != k23_binop_2(A,1) & B != k23_binop_2(A,2) & B != k23_binop_2(A,3) & B != k23_binop_2(A,4) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_jordan2c,t27_nat_1]), [file(jordan2c,t4_jordan2c),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(t20_sprect_2,theorem,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => k4_finseq_1(k3_goboard1(A)) = k4_finseq_1(A) ) ), file(sprect_2,t20_sprect_2), [interesting(0.00)]). fof(d4_goboard1,definition,( ! [A] : ( m2_finseq_1(A,u1_struct_0(k15_euclid(2))) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ( B = k3_goboard1(A) <=> ( k3_finseq_1(B) = k3_finseq_1(A) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(B)) => k1_goboard1(B,C) = k22_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,C)) ) ) ) ) ) ) ), file(goboard1,d4_goboard1), [interesting(0.00)]). fof(t114_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k22_euclid(k37_pscomp_1(A)) = k21_pscomp_1(A) & k22_euclid(k36_pscomp_1(A)) = k21_pscomp_1(A) ) ) ), file(pscomp_1,t114_pscomp_1), [interesting(0.00)]). fof(t94_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( k22_euclid(k32_pscomp_1(A)) = k19_pscomp_1(A) & k22_euclid(k33_pscomp_1(A)) = k19_pscomp_1(A) ) ) ), file(pscomp_1,t94_pscomp_1), [interesting(0.00)]). fof(t76_card_2,theorem,( ! [A,B] : ( A != B => k4_card_1(k2_tarski(A,B)) = 2 ) ), file(card_2,t76_card_2), [interesting(0.00)]). fof(t24_goboard2,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k3_finseq_1(k3_goboard2(A)) = k4_card_1(k2_relat_1(k2_goboard1(A))) & k1_matrix_1(k3_goboard2(A)) = k4_card_1(k2_relat_1(k3_goboard1(A))) ) ) ), file(goboard2,t24_goboard2), [interesting(0.00)]). fof(d1_goboard2,definition,( ! [A] : ( ( v1_goboard1(A) & m2_finseq_1(A,k1_numbers) ) => ! [B] : ( ( v1_goboard1(B) & m2_finseq_1(B,k1_numbers) ) => ~ ( A != k1_xboole_0 & B != k1_xboole_0 & ~ ! [C] : ( ( v1_matrix_1(C) & m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(2)))) ) => ( C = k1_goboard2(A,B) <=> ( k3_finseq_1(C) = k3_finseq_1(A) & k1_matrix_1(C) = k3_finseq_1(B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C)) => k3_matrix_1(u1_struct_0(k15_euclid(2)),C,D,E) = k23_euclid(k1_goboard1(A,D),k1_goboard1(B,E)) ) ) ) ) ) ) ) ) ) ), file(goboard2,d1_goboard2), [interesting(0.00)]). fof(d2_goboard2,definition,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ! [B] : ( ( v1_goboard1(B) & m2_finseq_1(B,k1_numbers) ) => ( B = k2_goboard2(A) <=> ( k2_relat_1(B) = k2_relat_1(A) & k3_finseq_1(B) = k4_card_1(k2_relat_1(A)) ) ) ) ) ), file(goboard2,d2_goboard2), [interesting(0.00)]). fof(d4_matrix_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_matrix_1(A) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( ~ r1_xreal_0(k3_finseq_1(A),0) => ( B = k1_matrix_1(A) <=> ? [C] : ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) & r2_hidden(C,k2_relat_1(A)) & k3_finseq_1(C) = B ) ) ) & ( r1_xreal_0(k3_finseq_1(A),0) => ( B = k1_matrix_1(A) <=> B = 0 ) ) ) ) ) ), file(matrix_1,d4_matrix_1), [interesting(0.00)]). fof(d1_matrix_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_matrix_1(A) <=> ? [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) & ! [C] : ~ ( r2_hidden(C,k2_relat_1(A)) & ! [D] : ( ( v1_relat_1(D) & v1_funct_1(D) & v1_finseq_1(D) ) => ~ ( D = C & k3_finseq_1(D) = B ) ) ) ) ) ) ), file(matrix_1,d1_matrix_1), [interesting(0.00)]). fof(d3_matrix_1,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( ( v1_matrix_1(D) & m2_finseq_1(D,k3_finseq_2(A)) ) => ( m1_matrix_1(D,A,B,C) <=> ( k3_finseq_1(D) = B & ! [E] : ( m2_finseq_1(E,A) => ( r2_hidden(E,k2_relat_1(D)) => k3_finseq_1(E) = C ) ) ) ) ) ) ) ) ), file(matrix_1,d3_matrix_1), [interesting(0.00)]). fof(t106_zfmisc_1,theorem,( ! [A,B,C,D] : ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D)) <=> ( r2_hidden(A,C) & r2_hidden(B,D) ) ) ), file(zfmisc_1,t106_zfmisc_1), [interesting(0.00)]). fof(t25_matrix_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_matrix_1(C,B,A,A) => ( k3_finseq_1(C) = A & k1_matrix_1(C) = A & k2_matrix_1(C) = k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(A)) ) ) ) ) ), file(matrix_1,t25_matrix_1), [interesting(0.00)]). fof(d1_goboard1,definition,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ( v1_goboard1(A) <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r2_hidden(B,k4_finseq_1(A)) & r2_hidden(C,k4_finseq_1(A)) & ~ r1_xreal_0(C,B) & r1_xreal_0(k1_goboard1(A,C),k1_goboard1(A,B)) ) ) ) ) ) ), file(goboard1,d1_goboard1), [interesting(0.00)]). fof(t7_jordan2c,theorem,( ! [A,B,C] : ( ( v1_relat_1(C) & v1_funct_1(C) & v1_finseq_1(C) ) => ~ ( k2_relat_1(C) = k2_tarski(A,B) & k3_finseq_1(C) = 2 & ~ ( k1_funct_1(C,1) = A & k1_funct_1(C,2) = B ) & ~ ( k1_funct_1(C,1) = B & k1_funct_1(C,2) = A ) ) ) ), inference(mizar_proof,[status(thm)],[t3_finseq_1,d3_finseq_1,d5_funct_1,t3_finseq_1,d3_finseq_1,d5_funct_1,d2_tarski,d5_funct_1,d3_finseq_1,t3_finseq_1,t27_nat_1,d2_tarski,d5_funct_1,d3_finseq_1,t3_finseq_1,t27_nat_1,d2_tarski]), [file(jordan2c,t7_jordan2c),interesting(0.00)]). fof(t8_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( ( v1_goboard1(C) & m2_finseq_1(C,k1_numbers) ) => ( ( k2_relat_1(C) = k2_tarski(A,B) & k3_finseq_1(C) = 2 & r1_xreal_0(A,B) ) => ( k1_goboard1(C,1) = A & k1_goboard1(C,2) = B ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,d1_goboard1,t7_jordan2c]), [file(jordan2c,t8_jordan2c),interesting(0.00)]). fof(t85_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k33_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k24_pscomp_1(A) ) ), file(sprect_1,t85_sprect_1), [interesting(0.00)]). fof(t76_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k21_euclid(k24_pscomp_1(A)) = k20_pscomp_1(A) ) ), file(pscomp_1,t76_pscomp_1), [interesting(0.00)]). fof(t69_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k20_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k20_pscomp_1(A) ) ), file(sprect_1,t69_sprect_1), [interesting(0.00)]). fof(t89_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k36_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k25_pscomp_1(A) ) ), file(sprect_1,t89_sprect_1), [interesting(0.00)]). fof(t78_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k21_euclid(k25_pscomp_1(A)) = k20_pscomp_1(A) ) ), file(pscomp_1,t78_pscomp_1), [interesting(0.00)]). fof(t88_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k37_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k22_pscomp_1(A) ) ), file(sprect_1,t88_sprect_1), [interesting(0.00)]). fof(t72_pscomp_1,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => k21_euclid(k22_pscomp_1(A)) = k18_pscomp_1(A) ) ), file(pscomp_1,t72_pscomp_1), [interesting(0.00)]). fof(t66_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k18_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k18_pscomp_1(A) ) ), file(sprect_1,t66_sprect_1), [interesting(0.00)]). fof(t97_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & ~ v1_sppol_1(A) & ~ v2_sppol_1(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => ( k3_finseq_1(k3_goboard2(k1_sprect_1(A))) = 2 & k1_matrix_1(k3_goboard2(k1_sprect_1(A))) = 2 & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),k1_sprect_1(A),1) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(k1_sprect_1(A)),1,2) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),k1_sprect_1(A),2) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(k1_sprect_1(A)),2,2) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),k1_sprect_1(A),3) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(k1_sprect_1(A)),2,1) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),k1_sprect_1(A),4) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(k1_sprect_1(A)),1,1) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),k1_sprect_1(A),5) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(k1_sprect_1(A)),1,2) ) ) ), inference(mizar_proof,[status(thm)],[t90_sprect_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t3_finseq_1,d3_finseq_1,t33_sprect_1,t34_sprect_1,t91_sprect_1,t92_sprect_1,t93_sprect_1,t94_sprect_1,d1_finseq_6,d3_goboard2,t19_sprect_2,d3_goboard1,t84_pscomp_1,t19_sprect_2,d3_goboard1,t104_pscomp_1,t19_sprect_2,d3_goboard1,t104_pscomp_1,t19_sprect_2,d3_goboard1,t84_pscomp_1,t19_sprect_2,d3_goboard1,t84_pscomp_1,d3_tarski,d5_funct_1,t19_sprect_2,d3_finseq_1,t3_finseq_1,t4_jordan2c,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d3_tarski,t19_sprect_2,d5_funct_1,t19_sprect_2,d5_funct_1,d2_tarski,d10_xboole_0,d2_goboard2,t20_sprect_2,d4_goboard1,t114_pscomp_1,t20_sprect_2,d4_goboard1,t94_pscomp_1,t20_sprect_2,d4_goboard1,t94_pscomp_1,t20_sprect_2,d4_goboard1,t114_pscomp_1,t20_sprect_2,d4_goboard1,t94_pscomp_1,d3_tarski,d5_funct_1,t20_sprect_2,d3_finseq_1,t3_finseq_1,t4_jordan2c,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d2_tarski,d3_tarski,t20_sprect_2,d5_funct_1,t20_sprect_2,d5_funct_1,d2_tarski,d10_xboole_0,d2_goboard2,t76_card_2,d2_goboard2,t76_card_2,d2_goboard2,t24_goboard2,t76_card_2,t24_goboard2,t76_card_2,t3_finseq_1,t3_finseq_1,d1_goboard2,d2_goboard2,d4_matrix_1,d3_goboard2,d1_matrix_1,d3_goboard2,d3_matrix_1,t3_finseq_1,t106_zfmisc_1,t3_finseq_1,t106_zfmisc_1,t25_matrix_1,t106_zfmisc_1,t25_matrix_1,t106_zfmisc_1,t25_matrix_1,t8_jordan2c,t84_pscomp_1,t56_euclid,d1_goboard2,t85_sprect_1,t76_pscomp_1,t69_sprect_1,t8_jordan2c,t56_euclid,d1_goboard2,t89_sprect_1,t78_pscomp_1,t69_sprect_1,t8_jordan2c,t56_euclid,d1_goboard2,t88_sprect_1,t72_pscomp_1,t66_sprect_1,t8_jordan2c,t56_euclid,d1_goboard2,t8_jordan2c,t94_pscomp_1,t56_euclid,d1_goboard2,t94_pscomp_1,t56_euclid,d1_goboard2,t8_jordan2c,t114_pscomp_1,t56_euclid,d1_goboard2,t114_pscomp_1,t56_euclid,d1_goboard2,t57_euclid,t57_euclid,t57_euclid,t57_euclid,t57_euclid]), [file(jordan2c,t97_jordan2c),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(t80_card_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(A,B) => r1_xreal_0(k4_card_1(A),k4_card_1(B)) ) ) ) ), file(card_1,t80_card_1), [interesting(0.00)]). fof(t54_card_2,theorem,( ! [A,B] : ( v1_finset_1(B) => ( ~ r2_hidden(A,B) => k4_card_1(k2_xboole_0(B,k1_tarski(A))) = k1_nat_1(k4_card_1(B),1) ) ) ), file(card_2,t54_card_2), [interesting(0.00)]). fof(t79_card_1,theorem,( ! [A] : k4_card_1(k1_tarski(A)) = 1 ), file(card_1,t79_card_1), [interesting(0.00)]). fof(d7_goboard5,definition,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(k1_nat_1(B,1),k3_finseq_1(A)) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) => ( C = k5_goboard5(A,B) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ~ ( r2_hidden(k4_tarski(D,E),k2_matrix_1(k3_goboard2(A))) & r2_hidden(k4_tarski(F,G),k2_matrix_1(k3_goboard2(A))) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(A),D,E) & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,k1_nat_1(B,1)) = k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(A),F,G) & ~ ( D = F & k1_nat_1(E,1) = G & C = k3_goboard5(k3_goboard2(A),k5_binarith(D,1),E) ) & ~ ( k1_nat_1(D,1) = F & E = G & C = k3_goboard5(k3_goboard2(A),D,E) ) & ~ ( D = k1_nat_1(F,1) & E = G & C = k3_goboard5(k3_goboard2(A),F,k5_binarith(G,1)) ) & ~ ( D = F & E = k1_nat_1(G,1) & C = k3_goboard5(k3_goboard2(A),D,G) ) ) ) ) ) ) ) ) ) ) ) ), file(goboard5,d7_goboard5), [interesting(0.00)]). fof(t28_goboard6,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( ~ v3_relat_1(B) & v1_matrix_1(B) & v3_goboard1(B) & v4_goboard1(B) & v5_goboard1(B) & v6_goboard1(B) & m2_finseq_1(B,k3_finseq_2(u1_struct_0(k15_euclid(2)))) ) => ( r1_xreal_0(1,A) => ( r1_xreal_0(k3_finseq_1(B),A) | k1_tops_1(k15_euclid(2),k3_goboard5(B,A,k1_matrix_1(B))) = a_2_5_goboard6(A,B) ) ) ) ) ), file(goboard6,t28_goboard6), [interesting(0.00)]). fof(t60_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k17_euclid(2,k23_euclid(A,B),k23_euclid(C,D)) = k23_euclid(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ), file(euclid,t60_euclid), [interesting(0.00)]). fof(t62_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => k18_euclid(A,2,k23_euclid(B,C)) = k23_euclid(k3_xcmplx_0(A,B),k3_xcmplx_0(A,C)) ) ) ) ), file(euclid,t62_euclid), [interesting(0.00)]). fof(t7_jordan5d,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) & r1_xreal_0(1,C) & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) ) => ( r1_xreal_0(k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(A),1,C)),k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B))) & r1_xreal_0(k21_euclid(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,B)),k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(2)),k3_goboard2(A),k3_finseq_1(k3_goboard2(A)),C))) ) ) ) ) ) ), file(jordan5d,t7_jordan5d), [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(t15_jordan2c,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [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 & v2_connsp_1(B,A) ) => v2_connsp_1(D,k3_pre_topc(A,C)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_connsp_1,t1_jordan1,t47_jordan6,d3_connsp_1]), [file(jordan2c,t15_jordan2c),interesting(0.00)]). fof(t38_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( v2_connsp_1(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r3_connsp_1(A,C) & ~ r1_xboole_0(B,C) & ~ r1_tarski(B,C) ) ) ) ) ) ), file(connsp_1,t38_connsp_1), [interesting(0.00)]). fof(t2_connsp_1,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( r1_connsp_1(A,B,C) => r1_xboole_0(B,C) ) ) ) ) ), file(connsp_1,t2_connsp_1), [interesting(0.00)]). fof(t100_jordan2c,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))) => ( ( r3_connsp_1(A,B) & r3_connsp_1(A,C) & v2_connsp_1(D,A) ) => ( r1_xboole_0(B,D) | r1_xboole_0(C,D) | B = C ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t38_connsp_1,t38_connsp_1,t36_connsp_1,d7_xboole_0,d7_xboole_0,t19_xboole_1,t3_xboole_1,d7_xboole_0,t2_connsp_1]), [file(jordan2c,t100_jordan2c),interesting(0.00)]). fof(d3_jordan2c,definition,( ! [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)))) => ( r1_jordan2c(A,B,C) <=> ( r4_connsp_1(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B),C) & v1_jordan2c(C,A) ) ) ) ) ) ), file(jordan2c,d3_jordan2c), [interesting(0.00)]). fof(t17_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)))) => ( r1_jordan2c(A,B,C) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B))))) & D = C & r3_connsp_1(k3_pre_topc(k15_euclid(A),k3_subset_1(u1_struct_0(k15_euclid(A)),B)),D) & v6_tbsp_1(D,k14_euclid(A)) & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d6_connsp_1,d3_jordan2c,d3_jordan2c,d2_jordan2c,d2_jordan2c,d6_connsp_1,d3_jordan2c]), [file(jordan2c,t17_jordan2c),interesting(0.00)]). fof(d2_connsp_3,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( m1_connsp_3(B,A) <=> ? [C] : ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) & ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) => ( r2_hidden(D,C) => r3_connsp_1(A,D) ) ) & B = k5_setfam_1(u1_struct_0(A),C) ) ) ) ) ), file(connsp_3,d2_connsp_3), [interesting(0.00)]). fof(t37_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))) => ~ ( r3_connsp_1(A,B) & r3_connsp_1(A,C) & B != C & ~ r1_xboole_0(B,C) ) ) ) ) ), file(connsp_1,t37_connsp_1), [interesting(0.00)]). fof(d7_metric_1,definition,( ! [A] : ( l1_metric_1(A) => ( v6_metric_1(A) <=> v2_metric_1(u1_metric_1(A),u1_struct_0(A)) ) ) ), file(metric_1,d7_metric_1), [interesting(0.00)]). fof(d3_metric_1,definition,( ! [A,B] : ( ( v1_funct_1(B) & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v2_metric_1(B,A) <=> ! [C] : ( m1_subset_1(C,A) => k1_metric_1(A,A,B,C,C) = 0 ) ) ) ), file(metric_1,d3_metric_1), [interesting(0.00)]). fof(d1_metric_1,definition,( ! [A] : ( l1_metric_1(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( m1_subset_1(C,u1_struct_0(A)) => k2_metric_1(A,B,C) = k1_metric_1(u1_struct_0(A),u1_struct_0(A),u1_metric_1(A),B,C) ) ) ) ), file(metric_1,d1_metric_1), [interesting(0.00)]). fof(t6_goboard6,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k14_euclid(A))) => ( C = k9_metric_1(k14_euclid(A),D,B) => v3_pre_topc(C,k15_euclid(A)) ) ) ) ) ) ), file(goboard6,t6_goboard6), [interesting(0.00)]). fof(t111_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k14_euclid(A))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k15_euclid(A))) => ( ( D = C & r2_hidden(E,k9_metric_1(k14_euclid(A),C,B)) ) => ( ~ r1_xreal_0(B,k5_toprns_1(A,k20_euclid(A,D,E))) & ~ r1_xreal_0(B,k5_toprns_1(A,k20_euclid(A,E,D))) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_topreal3,t12_metric_1,t45_jgraph_1]), [file(jordan2c,t111_jordan2c),interesting(0.00)]). fof(t34_connsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ~ ( r3_connsp_1(A,B) & B = k1_pre_topc(A) ) ) ) ), file(connsp_1,t34_connsp_1), [interesting(0.00)]). fof(t7_sprect_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))) => ( r4_connsp_1(A,C,B) => r1_tarski(B,C) ) ) ) ) ), file(sprect_1,t7_sprect_1), [interesting(0.00)]). fof(t8_xboole_1,theorem,( ! [A,B,C] : ( ( r1_tarski(A,B) & r1_tarski(C,B) ) => r1_tarski(k2_xboole_0(A,C),B) ) ), file(xboole_1,t8_xboole_1), [interesting(0.00)]). fof(t31_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)))) => k4_subset_1(u1_struct_0(k15_euclid(A)),k1_jordan2c(A,B),k2_jordan2c(A,B)) = k3_subset_1(u1_struct_0(k15_euclid(A)),B) ) ) ), inference(mizar_proof,[status(thm)],[t29_jordan2c,t30_jordan2c,t8_xboole_1,d3_tarski,t1_jordan1,t43_connsp_1,t12_pre_topc,d10_pre_topc,t1_xboole_1,t40_connsp_1,d6_connsp_1,d3_jordan2c,t26_jordan2c,d2_xboole_0,d4_jordan2c,t27_jordan2c,d2_xboole_0,d10_xboole_0]), [file(jordan2c,t31_jordan2c),interesting(0.00)]). fof(t115_jordan2c,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [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(A))) => ( ( r3_connsp_1(A,B) & r3_connsp_1(A,C) & r3_connsp_1(A,D) & k4_subset_1(u1_struct_0(A),B,C) = u1_struct_0(A) & r1_xboole_0(D,B) ) => D = C ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d7_xboole_0,t28_xboole_1,t23_xboole_1,t34_connsp_1,d7_xboole_0,t37_connsp_1]), [file(jordan2c,t115_jordan2c),interesting(0.00)]). fof(t62_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) => ~ ( r1_xreal_0(1,B) & C = k4_xboole_0(k1_euclid(B),a_2_4_jordan2c(A,B)) & v1_jordan2c(C,B) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t40_jordan2c,t60_jordan2c,t26_toprns_1,d4_xboole_0,t25_euclid,t8_toprns_1,t37_jordan2c,t132_complex1,t94_square_1,t66_xreal_1,t83_square_1,t11_absvalue,t2_xreal_1,t31_xreal_1,t31_xreal_1,t2_xreal_1,t2_xreal_1,t8_toprns_1,t37_jordan2c,t132_complex1,t94_square_1,t66_xreal_1,t83_square_1,t11_absvalue,t2_xreal_1,t31_xreal_1,t2_xreal_1,t31_xreal_1,t2_xreal_1,d3_tarski,t26_toprns_1,t25_euclid,t12_pre_topc,t41_jordan2c]), [file(jordan2c,t62_jordan2c),interesting(0.00)]). fof(t24_jordan2b,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(1))) => ? [B] : ( m1_subset_1(B,k1_numbers) & A = k13_binarith(k1_numbers,B) ) ) ), file(jordan2b,t24_jordan2b), [interesting(0.00)]). fof(d2_jordan2b,definition,( ! [A] : ( v1_xreal_0(A) => k2_jordan2b(A) = k9_finseq_1(A) ) ), file(jordan2b,d2_jordan2b), [interesting(0.00)]). fof(t116_rvsum_1,theorem,( ! [A] : ( m2_finseq_1(A,k1_numbers) => r1_xreal_0(0,k15_rvsum_1(k11_rvsum_1(A))) ) ), file(rvsum_1,t116_rvsum_1), [interesting(0.00)]). fof(d4_square_1,definition,( ! [A] : ( v1_xreal_0(A) => ( r1_xreal_0(0,A) => ! [B] : ( v1_xreal_0(B) => ( B = k8_square_1(A) <=> ( r1_xreal_0(0,B) & k5_square_1(B) = A ) ) ) ) ) ), file(square_1,d4_square_1), [interesting(0.00)]). fof(t81_rvsum_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => k11_rvsum_1(k12_finseq_1(k1_numbers,A)) = k12_finseq_1(k1_numbers,k7_square_1(A)) ) ), file(rvsum_1,t81_rvsum_1), [interesting(0.00)]). fof(t103_rvsum_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => k15_rvsum_1(k12_finseq_1(k1_numbers,A)) = A ) ), file(rvsum_1,t103_rvsum_1), [interesting(0.00)]). fof(t158_complex1,theorem,( ! [A] : ( v1_xreal_0(A) => k8_square_1(k5_square_1(A)) = k17_complex1(A) ) ), file(complex1,t158_complex1), [interesting(0.00)]). fof(t29_jordan2b,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( k2_jordan2b(A) = k2_jordan2b(B) => A = B ) ) ) ), file(jordan2b,t29_jordan2b), [interesting(0.00)]). fof(t9_seq_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ~ r1_xreal_0(B,k4_xcmplx_0(A)) & ~ r1_xreal_0(A,B) ) <=> ~ r1_xreal_0(A,k18_complex1(B)) ) ) ) ), file(seq_2,t9_seq_2), [interesting(0.00)]). fof(t26_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k4_xcmplx_0(B),k4_xcmplx_0(A)) ) ) ) ), file(xreal_1,t26_xreal_1), [interesting(0.00)]). fof(t7_absvalue,theorem,( ! [A] : ( v1_xreal_0(A) => ( A = 0 <=> k18_complex1(A) = 0 ) ) ), file(absvalue,t7_absvalue), [interesting(0.00)]). fof(t109_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,k4_xcmplx_0(B)) => ( r1_xreal_0(k2_xcmplx_0(A,B),0) & r1_xreal_0(B,k4_xcmplx_0(A)) ) ) ) ) ), file(real_2,t109_real_2), [interesting(0.00)]). fof(t26_jordan2b,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m1_subset_1(B,k1_numbers) => k18_euclid(B,1,k2_jordan2b(A)) = k2_jordan2b(k3_xcmplx_0(B,A)) ) ) ), file(jordan2b,t26_jordan2b), [interesting(0.00)]). fof(t27_jordan2b,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k2_jordan2b(k2_xcmplx_0(A,B)) = k17_euclid(1,k2_jordan2b(A),k2_jordan2b(B)) ) ) ), file(jordan2b,t27_jordan2b), [interesting(0.00)]). fof(t173_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( ~ r1_xreal_0(0,A) => ( ~ r1_xreal_0(B,k2_xcmplx_0(A,B)) & ~ r1_xreal_0(k6_xcmplx_0(B,A),B) ) ) & ~ ( ~ ( r1_xreal_0(B,k2_xcmplx_0(A,B)) & r1_xreal_0(k6_xcmplx_0(B,A),B) ) & r1_xreal_0(0,A) ) ) ) ) ), file(real_2,t173_real_2), [interesting(0.00)]). fof(t138_complex1,theorem,( ! [A] : ( v1_xcmplx_0(A) => k17_complex1(k4_xcmplx_0(A)) = k17_complex1(A) ) ), file(complex1,t138_complex1), [interesting(0.00)]). fof(t18_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_1), [interesting(0.00)]). fof(t112_finseq_2,theorem,( ! [A] : k4_finseq_2(0,A) = k6_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,A)),k6_finseq_1(A)) ), file(finseq_2,t112_finseq_2), [interesting(0.00)]). fof(t72_finseq_2,theorem,( ! [A] : k2_finseq_2(0,A) = k1_xboole_0 ), file(finseq_2,t72_finseq_2), [interesting(0.00)]). fof(t57_sprect_3,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_sprect_1(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( ~ ( r1_xreal_0(k18_pscomp_1(k5_topreal1(2,A)),k21_euclid(B)) & r1_xreal_0(k21_euclid(B),k20_pscomp_1(k5_topreal1(2,A))) & r1_xreal_0(k21_pscomp_1(k5_topreal1(2,A)),k22_euclid(B)) & r1_xreal_0(k22_euclid(B),k19_pscomp_1(k5_topreal1(2,A))) ) => r2_hidden(B,k2_goboard9(A)) ) ) ) ), file(sprect_3,t57_sprect_3), [interesting(0.00)]). fof(t58_sprect_3,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_sprect_2(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) => r1_tarski(k2_goboard9(k1_sprect_1(k5_topreal1(2,A))),k2_goboard9(A)) ) ) ), file(sprect_3,t58_sprect_3), [interesting(0.00)]). fof(t118_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & v1_sprect_2(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) => ( r1_xreal_0(k18_pscomp_1(k5_topreal1(2,A)),k21_euclid(B)) | r2_hidden(B,k2_goboard9(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t66_sprect_1,t57_sprect_3,t58_sprect_3]), [file(jordan2c,t118_jordan2c),interesting(0.00)]). fof(t119_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & v1_sprect_2(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) => ( r1_xreal_0(k21_euclid(B),k20_pscomp_1(k5_topreal1(2,A))) | r2_hidden(B,k2_goboard9(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t69_sprect_1,t57_sprect_3,t58_sprect_3]), [file(jordan2c,t119_jordan2c),interesting(0.00)]). fof(t67_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k21_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k21_pscomp_1(A) ) ), file(sprect_1,t67_sprect_1), [interesting(0.00)]). fof(t120_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & v1_sprect_2(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) => ( r1_xreal_0(k21_pscomp_1(k5_topreal1(2,A)),k22_euclid(B)) | r2_hidden(B,k2_goboard9(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t67_sprect_1,t57_sprect_3,t58_sprect_3]), [file(jordan2c,t120_jordan2c),interesting(0.00)]). fof(t68_sprect_1,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v6_compts_1(A,k15_euclid(2)) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(2)))) ) => k19_pscomp_1(k5_topreal1(2,k1_sprect_1(A))) = k19_pscomp_1(A) ) ), file(sprect_1,t68_sprect_1), [interesting(0.00)]). fof(t121_jordan2c,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & ~ v5_seqm_3(A) & v1_topreal1(A) & v2_topreal1(A) & v1_finseq_6(A,u1_struct_0(k15_euclid(2))) & v1_goboard5(A) & v2_goboard5(A) & v1_sprect_2(A) & m2_finseq_1(A,u1_struct_0(k15_euclid(2))) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => ( k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(2)),A,1) = k32_pscomp_1(k5_topreal1(2,A)) => ( r1_xreal_0(k22_euclid(B),k19_pscomp_1(k5_topreal1(2,A))) | r2_hidden(B,k2_goboard9(A)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t68_sprect_1,t57_sprect_3,t58_sprect_3]), [file(jordan2c,t121_jordan2c),interesting(0.00)]). fof(t19_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)))) => ( r1_jordan2c(A,B,C) => r1_tarski(C,k3_subset_1(u1_struct_0(k15_euclid(A)),B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_jordan2c,t7_sprect_1]), [file(jordan2c,t19_jordan2c),interesting(0.00)]). fof(t20_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)))) => ( r2_jordan2c(A,B,C) => r1_tarski(C,k3_subset_1(u1_struct_0(k15_euclid(A)),B)) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d4_jordan2c,t7_sprect_1]), [file(jordan2c,t20_jordan2c),interesting(0.00)]). fof(t17_finseqop,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,A) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,A,B) & m2_relset_1(E,A,B) ) => k5_finseqop(A,B,k4_finseqop(A,D,C),E) = k4_finseqop(B,D,k8_funct_2(A,B,E,C)) ) ) ) ) ) ), file(finseqop,t17_finseqop), [interesting(0.00)]). fof(d2_euclid,definition,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k1_numbers,k1_numbers) & m2_relset_1(A,k1_numbers,k1_numbers) ) => ( A = k2_euclid <=> ! [B] : ( v1_xreal_0(B) => k1_funct_1(A,B) = k18_complex1(B) ) ) ) ), file(euclid,d2_euclid), [interesting(0.00)]). fof(t51_jordan2c,theorem,( $true ), file(jordan2c,t51_jordan2c), [interesting(0.00)]). fof(t5_jordan2c,theorem,( $true ), file(jordan2c,t5_jordan2c), [interesting(0.00)]). fof(t6_jordan2c,theorem,( $true ), file(jordan2c,t6_jordan2c), [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(t10_compts_1,theorem,( ! [A] : ( l1_pre_topc(A) => ( v2_compts_1(A) <=> v6_compts_1(k2_pre_topc(A),A) ) ) ), file(compts_1,t10_compts_1), [interesting(0.00)]). fof(t20_topmetr,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] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) => ( B = C => k3_pre_topc(k15_euclid(A),B) = k5_pcomps_1(k1_topmetr(k14_euclid(A),C)) ) ) ) ) ), file(topmetr,t20_topmetr), [interesting(0.00)]). fof(t12_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( ( v6_metric_1(A) & v9_metric_1(A) & v2_compts_1(k5_pcomps_1(A)) ) => v1_tbsp_1(A) ) ) ), file(tbsp_1,t12_tbsp_1), [interesting(0.00)]). fof(t26_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ( v1_tbsp_1(A) => v5_tbsp_1(A) ) ) ), file(tbsp_1,t26_tbsp_1), [interesting(0.00)]). fof(t25_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( v5_tbsp_1(A) <=> v6_tbsp_1(k2_pre_topc(A),A) ) ) ), file(tbsp_1,t25_tbsp_1), [interesting(0.00)]). fof(d2_topmetr,definition,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( ( v1_metric_1(C) & m1_topmetr(C,A) ) => ( C = k1_topmetr(A,B) <=> u1_struct_0(C) = B ) ) ) ) ), file(topmetr,d2_topmetr), [interesting(0.00)]). fof(d1_topmetr,definition,( ! [A] : ( ( v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) & l1_metric_1(B) ) => ( m1_topmetr(B,A) <=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A)) & ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => ! [D] : ( m1_subset_1(D,u1_struct_0(B)) => k1_metric_1(u1_struct_0(B),u1_struct_0(B),u1_metric_1(B),C,D) = k1_binop_1(u1_metric_1(A),C,D) ) ) ) ) ) ) ), file(topmetr,d1_topmetr), [interesting(0.00)]). fof(t72_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) => ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) ) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k1_topmetr(k14_euclid(A),C)))) => ( ( r1_tarski(B,C) & B = D & v6_tbsp_1(D,k1_topmetr(k14_euclid(A),C)) ) => v6_tbsp_1(B,k14_euclid(A)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d9_tbsp_1,d1_topmetr,d1_metric_1,d1_metric_1,d9_tbsp_1]), [file(jordan2c,t72_jordan2c),interesting(0.00)]). fof(t13_connsp_3,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v2_connsp_1(B,A) & v2_connsp_1(C,A) & r1_tarski(B,C) ) => ( B = k1_xboole_0 | r1_tarski(C,k1_connsp_3(A,B)) ) ) ) ) ) ), file(connsp_3,t13_connsp_3), [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(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(t33_toprns_1,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))) => r1_xreal_0(k5_real_1(k5_toprns_1(A,B),k5_toprns_1(A,C)),k5_toprns_1(A,k20_euclid(A,B,C))) ) ) ) ), file(toprns_1,t33_toprns_1), [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(t88_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] : ( v1_xreal_0(C) => ( B = a_2_5_jordan2c(A,C) => ( v3_pre_topc(k3_subset_1(u1_struct_0(k15_euclid(A)),B),k15_euclid(A)) & v4_pre_topc(B,k15_euclid(A)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d1_xreal_0,d5_subset_1,d4_xboole_0,t13_topreal3,d5_real_1,t52_xreal_1,t127_real_2,d3_tarski,t13_topreal3,t12_metric_1,t45_jgraph_1,t33_toprns_1,t2_xreal_1,t22_xreal_1,t127_real_2,d4_xboole_0,d5_subset_1,t52_xreal_1,t127_real_2,d3_tarski,t13_topreal3,t12_metric_1,t45_jgraph_1,t33_toprns_1,t2_xreal_1,t22_xreal_1,t127_real_2,d4_xboole_0,d5_subset_1,t22_topmetr,t29_tops_1]), [file(jordan2c,t88_jordan2c),interesting(0.00)]). fof(t32_tops_2,theorem,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) => ( v3_pre_topc(C,B) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & v3_pre_topc(D,A) & k3_xboole_0(D,k2_pre_topc(B)) = C ) ) ) ) ) ), file(tops_2,t32_tops_2), [interesting(0.00)]). fof(t38_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))) => ( ( v3_pre_topc(B,A) & v3_pre_topc(C,A) ) => v3_pre_topc(k5_subset_1(u1_struct_0(A),B,C),A) ) ) ) ) ), file(tops_1,t38_tops_1), [interesting(0.00)]). fof(t18_metric_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( ( ~ v3_struct_0(B) & l1_metric_1(B) ) => ! [C] : ( m1_subset_1(C,u1_struct_0(B)) => k9_metric_1(B,C,A) = a_3_0_metric_1(A,B,C) ) ) ) ), file(metric_1,t18_metric_1), [interesting(0.00)]). fof(t78_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k14_euclid(A))) => ! [C] : ( m1_subset_1(C,k1_numbers) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( D = k9_metric_1(k14_euclid(A),B,C) => v1_jordan1(D,A) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t18_metric_1,t13_topreal3,t45_jgraph_1,t45_jgraph_1,t13_topreal3,t45_jgraph_1,d3_tarski,d3_topreal1,t37_euclid,t33_euclid,t50_euclid,t45_euclid,t45_euclid,t30_euclid,t30_euclid,t44_euclid,t36_euclid,t45_euclid,t44_euclid,t36_euclid,t45_euclid,t30_toprns_1,t8_toprns_1,t50_xreal_1,d1_absvalue,t70_xreal_1,t8_toprns_1,t132_complex1,t66_xreal_1,d1_absvalue,t10_xreal_1,t2_xreal_1,t1_xreal_1,t66_xreal_1,t8_toprns_1,d1_absvalue,t70_xreal_1,d1_absvalue,t10_xreal_1,t2_xreal_1,d1_jordan1]), [file(jordan2c,t78_jordan2c),interesting(0.00)]). fof(t4_goboard6,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( v1_xreal_0(C) => ( ~ r1_xreal_0(C,0) => r2_hidden(B,k9_metric_1(A,B,C)) ) ) ) ) ), file(goboard6,t4_goboard6), [interesting(0.00)]). fof(t24_connsp_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_connsp_2(A) <=> ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ( ( v3_pre_topc(B,A) & r4_connsp_1(A,B,C) ) => v3_pre_topc(C,A) ) ) ) ) ) ), file(connsp_2,t24_connsp_2), [interesting(0.00)]). fof(t90_jordan2c,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(k15_euclid(A)))) => ! [D] : ( v1_xreal_0(D) => ( ( C = a_2_5_jordan2c(A,D) & k3_subset_1(u1_struct_0(k15_euclid(A)),C) = B ) => v1_connsp_2(k3_pre_topc(k15_euclid(A),B)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t88_jordan2c,t89_jordan2c]), [file(jordan2c,t90_jordan2c),interesting(0.00)]). fof(t21_connsp_2,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_connsp_2(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ( r3_connsp_1(A,B) => v3_pre_topc(B,A) ) ) ) ) ), file(connsp_2,t21_connsp_2), [interesting(0.00)]). fof(t34_connsp_3,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 & v2_connsp_1(D,k3_pre_topc(A,C)) ) => v2_connsp_1(B,A) ) ) ) ) ) ), file(connsp_3,t34_connsp_3), [interesting(0.00)]). fof(t85_jordan2c,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( r2_hidden(B,D) & C = a_3_1_jordan2c(A,B,D) ) => r1_tarski(C,D) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t83_borsuk_1,d1_funct_2,t48_rcomp_1,d5_funct_1]), [file(jordan2c,t85_jordan2c),interesting(0.00)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.00)]). fof(t16_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(A)) => ! [C] : ( v1_xreal_0(C) => ( v6_metric_1(A) => ( r1_xreal_0(C,0) | r2_hidden(B,k9_metric_1(A,B,C)) ) ) ) ) ) ), file(tbsp_1,t16_tbsp_1), [interesting(0.00)]). fof(t4_jordan6,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_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,C))) & m2_relset_1(D,u1_struct_0(A),u1_struct_0(k3_pre_topc(B,C))) ) => ( 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(A),u1_struct_0(B)) & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) ) => ( ( E = D & v5_pre_topc(D,A,k3_pre_topc(B,C)) ) => v5_pre_topc(E,A,B) ) ) ) ) ) ) ) ), file(jordan6,t4_jordan6), [interesting(0.00)]). fof(t80_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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(k14_euclid(A))) => ~ ( C != D & r2_hidden(C,k9_metric_1(k14_euclid(A),E,B)) & r2_hidden(D,k9_metric_1(k14_euclid(A),E,B)) & ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(F,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(F,k5_topmetr,k15_euclid(A)) & k1_funct_1(F,0) = C & k1_funct_1(F,1) = D & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),F),k9_metric_1(k14_euclid(A),E,B)) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_topreal3,t78_jordan2c,d1_jordan1,t15_topreal1,d2_topreal1,d5_tops_2,t4_jordan6,t4_jordan6,d10_pre_topc]), [file(jordan2c,t80_jordan2c),interesting(0.00)]). fof(t28_topreal3,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(B))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(B))) => ! [E] : ( m1_subset_1(E,u1_struct_0(k14_euclid(B))) => ( ( r2_hidden(C,k9_metric_1(k14_euclid(B),E,A)) & r2_hidden(D,k9_metric_1(k14_euclid(B),E,A)) ) => r1_tarski(k3_topreal1(B,C,D),k9_metric_1(k14_euclid(B),E,A)) ) ) ) ) ) ) ), file(topreal3,t28_topreal3), [interesting(0.00)]). fof(t9_xboole_1,theorem,( ! [A,B,C] : ( r1_tarski(A,B) => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,C)) ) ), file(xboole_1,t9_xboole_1), [interesting(0.00)]). fof(t81_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k14_euclid(A))) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(G,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(G,k5_topmetr,k15_euclid(A)) & k1_funct_1(G,0) = C & k1_funct_1(G,1) = D & r2_hidden(E,k9_metric_1(k14_euclid(A),F,B)) & r2_hidden(D,k9_metric_1(k14_euclid(A),F,B)) & ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(H,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(H,k5_topmetr,k15_euclid(A)) & k1_funct_1(H,0) = C & k1_funct_1(H,1) = E & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),H),k2_xboole_0(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),G),k9_metric_1(k14_euclid(A),F,B))) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t28_topreal3,t15_topreal1,d2_topreal1,d5_tops_2,t4_jordan6,t4_jordan6,d10_pre_topc,t32_jordan2c,t9_xboole_1,t1_xboole_1,t7_xboole_1]), [file(jordan2c,t81_jordan2c),interesting(0.00)]). fof(t82_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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))) => ! [F] : ( m1_subset_1(F,u1_struct_0(k14_euclid(A))) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [H] : ( ( v1_funct_1(H) & v1_funct_2(H,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(H,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(H,k5_topmetr,k15_euclid(A)) & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),H),G) & k1_funct_1(H,0) = C & k1_funct_1(H,1) = D & r2_hidden(E,k9_metric_1(k14_euclid(A),F,B)) & r2_hidden(D,k9_metric_1(k14_euclid(A),F,B)) & r1_tarski(k9_metric_1(k14_euclid(A),F,B),G) & ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(I,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(I,k5_topmetr,k15_euclid(A)) & r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),I),G) & k1_funct_1(I,0) = C & k1_funct_1(I,1) = E ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t81_jordan2c,t8_xboole_1,t1_xboole_1]), [file(jordan2c,t82_jordan2c),interesting(0.00)]). fof(t83_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,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( v2_connsp_1(B,k15_euclid(A)) & v3_pre_topc(B,k15_euclid(A)) & D = a_3_0_jordan2c(A,B,C) ) => v3_pre_topc(D,k15_euclid(A)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[d3_tarski,t13_topreal3,t22_topmetr,d1_xreal_0,t16_tbsp_1,d3_tarski,t16_tbsp_1,t80_jordan2c,t1_xboole_1,t82_jordan2c,t22_topmetr]), [file(jordan2c,t83_jordan2c),interesting(0.00)]). fof(t84_jordan2c,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,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) => ( ( v2_connsp_1(C,k15_euclid(A)) & v3_pre_topc(C,k15_euclid(A)) & r2_hidden(B,C) & D = a_3_1_jordan2c(A,B,C) ) => v3_pre_topc(D,k15_euclid(A)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t13_topreal3,t83_borsuk_1,d1_funct_2,t48_rcomp_1,d5_funct_1,t22_topmetr,d1_xreal_0,t16_tbsp_1,d3_tarski,d3_tarski,t82_jordan2c,t16_tbsp_1,t80_jordan2c,t1_xboole_1,t22_topmetr]), [file(jordan2c,t84_jordan2c),interesting(0.00)]). fof(t16_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => ! [B] : ( r1_tarski(B,k2_pre_topc(A)) => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) ), file(pre_topc,t16_pre_topc), [interesting(0.00)]). fof(t39_xboole_1,theorem,( ! [A,B] : k2_xboole_0(A,k4_xboole_0(B,A)) = k2_xboole_0(A,B) ), file(xboole_1,t39_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(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(d9_pre_topc,definition,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( l1_pre_topc(B) => ( m1_pre_topc(B,A) <=> ( r1_tarski(k2_pre_topc(B),k2_pre_topc(A)) & ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) => ( r2_hidden(C,u1_pre_topc(B)) <=> ? [D] : ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A))) & r2_hidden(D,u1_pre_topc(A)) & C = k3_xboole_0(D,k2_pre_topc(B)) ) ) ) ) ) ) ) ), file(pre_topc,d9_pre_topc), [interesting(0.00)]). fof(t12_connsp_1,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ( v1_connsp_1(A) <=> ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) => ~ ( k2_pre_topc(A) = k4_subset_1(u1_struct_0(A),B,C) & B != k1_pre_topc(A) & C != k1_pre_topc(A) & v3_pre_topc(B,A) & v3_pre_topc(C,A) & r1_xboole_0(B,C) ) ) ) ) ) ), file(connsp_1,t12_connsp_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(t86_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)))) => ! [D] : ( m1_subset_1(D,u1_struct_0(k15_euclid(A))) => ( ( v2_connsp_1(C,k15_euclid(A)) & v3_pre_topc(C,k15_euclid(A)) & r2_hidden(D,C) & B = a_3_2_jordan2c(A,C,D) ) => r1_tarski(C,B) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t85_jordan2c,t36_xboole_1,d4_xboole_0,d4_xboole_0,t2_tarski,t83_jordan2c,t84_jordan2c,d3_connsp_1,d10_pre_topc,t16_pre_topc,t79_xboole_1,t39_xboole_1,t12_xboole_1,d5_pre_topc,t28_xboole_1,d9_pre_topc,d5_pre_topc,t12_connsp_1,t37_xboole_1]), [file(jordan2c,t86_jordan2c),interesting(0.00)]). fof(d10_jordan2c,definition,( ! [A] : ( ( ~ v3_struct_0(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)) ) => ! [D] : ( m1_subset_1(D,u1_struct_0(A)) => k6_jordan2c(A,B,C,D) = k1_funct_1(C,D) ) ) ) ) ), file(jordan2c,d10_jordan2c), [interesting(0.00)]). fof(s2_funct_1,theorem, ( ( ! [A,B,C] : ( ( r2_hidden(A,f1_s2_funct_1) & p1_s2_funct_1(A,B) & p1_s2_funct_1(A,C) ) => B = C ) & ! [A] : ~ ( r2_hidden(A,f1_s2_funct_1) & ! [B] : ~ p1_s2_funct_1(A,B) ) ) => ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & k1_relat_1(A) = f1_s2_funct_1 & ! [B] : ( r2_hidden(B,f1_s2_funct_1) => p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) ), file(funct_1,s2_funct_1), [interesting(0.00)]). fof(t24_topmetr,theorem,( u1_struct_0(k3_topmetr) = k1_numbers ), file(topmetr,t24_topmetr), [interesting(0.00)]). fof(d14_metric_1,definition,( k8_metric_1 = g1_metric_1(k1_numbers,k7_metric_1) ), file(metric_1,d14_metric_1), [interesting(0.00)]). fof(d13_metric_1,definition,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(A,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ) => ( A = k7_metric_1 <=> ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_numbers) => k1_metric_1(k1_numbers,k1_numbers,A,B,C) = k18_complex1(k5_real_1(B,C)) ) ) ) ) ), file(metric_1,d13_metric_1), [interesting(0.00)]). fof(t28_toprns_1,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))) => k5_toprns_1(A,k20_euclid(A,B,C)) = k5_toprns_1(A,k20_euclid(A,C,B)) ) ) ) ), file(toprns_1,t28_toprns_1), [interesting(0.00)]). fof(t13_uniform1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => k18_complex1(k5_real_1(A,B)) = k18_complex1(k5_real_1(B,A)) ) ) ), file(uniform1,t13_uniform1), [interesting(0.00)]). fof(t11_jordan2c,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))) => r1_xreal_0(k18_complex1(k10_binop_2(k5_toprns_1(A,B),k5_toprns_1(A,C))),k5_toprns_1(A,k20_euclid(A,B,C))) ) ) ) ), inference(mizar_proof,[status(thm)],[t50_xreal_1,d1_absvalue,t33_toprns_1,t52_xreal_1,t33_toprns_1,d1_absvalue,t28_toprns_1,t13_uniform1]), [file(jordan2c,t11_jordan2c),interesting(0.00)]). fof(t4_uniform1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) & l1_metric_1(A) ) => ! [B] : ( ( ~ v3_struct_0(B) & v6_metric_1(B) & v7_metric_1(B) & v8_metric_1(B) & v9_metric_1(B) & l1_metric_1(B) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) ) => ( ! [D] : ( v1_xreal_0(D) => ! [E] : ( m1_subset_1(E,u1_struct_0(A)) => ! [F] : ( m1_subset_1(F,u1_struct_0(B)) => ~ ( ~ r1_xreal_0(D,0) & F = k1_funct_1(C,E) & ! [G] : ( v1_xreal_0(G) => ~ ( ~ r1_xreal_0(G,0) & ! [H] : ( m1_subset_1(H,u1_struct_0(A)) => ! [I] : ( m1_subset_1(I,u1_struct_0(B)) => ~ ( I = k1_funct_1(C,H) & ~ r1_xreal_0(G,k4_metric_1(A,E,H)) & r1_xreal_0(D,k4_metric_1(B,F,I)) ) ) ) ) ) ) ) ) ) => v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B)) ) ) ) ) ), file(uniform1,t4_uniform1), [interesting(0.00)]). fof(t91_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) & m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) ) => ( ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),B,C) = k5_toprns_1(A,C) ) => v5_pre_topc(B,k15_euclid(A),k3_topmetr) ) ) ) ), inference(mizar_proof,[status(thm)],[d7_topmetr,t13_topreal3,t45_jgraph_1,d14_metric_1,d1_metric_1,d13_metric_1,d14_metric_1,t11_jordan2c,t2_xreal_1,d7_topmetr,t4_uniform1]), [file(jordan2c,t91_jordan2c),interesting(0.00)]). fof(t92_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) & m2_relset_1(B,u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr)) & ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k8_funct_2(u1_struct_0(k15_euclid(A)),u1_struct_0(k3_topmetr),B,C) = k5_toprns_1(A,C) ) & v5_pre_topc(B,k15_euclid(A),k3_topmetr) ) ) ), inference(mizar_proof,[status(thm)],[s2_funct_1,d3_tarski,d5_funct_1,t24_topmetr,d1_funct_2,t11_relset_1,t91_jordan2c]), [file(jordan2c,t92_jordan2c),interesting(0.00)]). fof(t93_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ~ ( v5_pre_topc(B,k5_topmetr,k15_euclid(A)) & ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_topmetr)) & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k3_topmetr)) ) => ~ ( ! [D] : ( m1_subset_1(D,u1_struct_0(k5_topmetr)) => k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k3_topmetr),C,D) = k5_toprns_1(A,k8_funct_2(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),B,D)) ) & v5_pre_topc(C,k5_topmetr,k3_topmetr) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t92_jordan2c,d1_funct_2,t23_funct_1,t58_tops_2]), [file(jordan2c,t93_jordan2c),interesting(0.00)]). fof(t12_topreal5,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) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) & v5_pre_topc(E,k4_topmetr(A,B),k3_topmetr) & m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) ) => ! [F] : ( v1_xreal_0(F) => ~ ( k1_funct_1(E,A) = C & k1_funct_1(E,B) = D & ~ r1_xreal_0(F,C) & ~ r1_xreal_0(D,F) & ! [G] : ( m1_subset_1(G,k1_numbers) => ~ ( k1_funct_1(E,G) = F & ~ r1_xreal_0(G,A) & ~ r1_xreal_0(B,G) ) ) ) ) ) ) ) ) ) ) ), file(topreal5,t12_topreal5), [interesting(0.00)]). fof(t94_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) ) => ! [C] : ( m1_subset_1(C,k1_numbers) => ~ ( v5_pre_topc(B,k5_topmetr,k15_euclid(A)) & r1_xreal_0(k5_toprns_1(A,k6_jordan2c(k5_topmetr,k15_euclid(A),B,0)),C) & r1_xreal_0(C,k5_toprns_1(A,k6_jordan2c(k5_topmetr,k15_euclid(A),B,1))) & ! [D] : ( m1_subset_1(D,u1_struct_0(k5_topmetr)) => k5_toprns_1(A,k6_jordan2c(k5_topmetr,k15_euclid(A),B,D)) != C ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t93_jordan2c,t48_rcomp_1,t83_borsuk_1,t48_rcomp_1,d10_jordan2c,t83_borsuk_1,t48_rcomp_1,t83_borsuk_1,t48_rcomp_1,d10_jordan2c,t27_topmetr,t12_topreal5,t83_borsuk_1,t48_rcomp_1,t83_borsuk_1,t48_rcomp_1,d10_jordan2c,t83_borsuk_1,d5_real_1]), [file(jordan2c,t94_jordan2c),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(t26_subset_1,theorem,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => r1_xboole_0(B,k3_subset_1(A,B)) ) ), file(subset_1,t26_subset_1), [interesting(0.00)]). fof(t49_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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))) => ~ ( C = a_2_2_jordan2c(A,B) & r2_hidden(D,C) & r2_hidden(E,C) & ! [F] : ( m1_subset_1(F,k1_numbers) => ( D != k18_euclid(F,A,E) & E != k18_euclid(F,A,D) ) ) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(F,C) & r2_hidden(G,C) & r1_tarski(k3_topreal1(A,D,F),C) & r1_tarski(k3_topreal1(A,F,G),C) & r1_tarski(k3_topreal1(A,G,E),C) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t47_jordan2c,t48_jordan2c,d3_tarski,d3_topreal1,t13_topreal3,d3_topreal1,d3_tarski,d12_funct_1,t13_topreal3,d6_weierstr,d12_funct_1,t13_topreal3,d6_weierstr,t5_metric_1,d2_seq_4,d1_funct_2,d6_weierstr,d12_funct_1,d5_seq_4,d3_weierstr,d5_weierstr,d8_weierstr,t45_jgraph_1,t13_jordan2c,t132_complex1,t26_toprns_1,t153_complex1,d1_absvalue,t143_real_2,t68_xreal_1,d9_xcmplx_0,d9_xcmplx_0,t31_xreal_1,t11_absvalue,t2_xreal_1,t2_xreal_1,t8_toprns_1,t36_euclid,t34_euclid,t34_euclid,t34_euclid,t34_euclid,t6_topreal1,d3_tarski,d3_topreal1,t31_xreal_1,t127_real_2,t121_real_2,t33_euclid,t31_euclid,t33_euclid,t13_topreal3,d3_topreal1,d3_tarski,d12_funct_1,t13_topreal3,d6_weierstr,d12_funct_1,t13_topreal3,d6_weierstr,t5_metric_1,d2_seq_4,d1_funct_2,d6_weierstr,d12_funct_1,d5_seq_4,d3_weierstr,d5_weierstr,d8_weierstr,t45_jgraph_1,t13_jordan2c,t75_xcmplx_1,t88_xcmplx_1,t66_xreal_1,t50_xreal_1,t8_xreal_1,t70_xreal_1,l5_jordan2c,d1_absvalue,t10_xreal_1,t2_xreal_1,t8_toprns_1,t37_euclid,t34_euclid,t8_xreal_1,t1_xreal_1,t33_euclid,t33_euclid,t31_euclid,t26_toprns_1,d3_tarski,d3_topreal1,t31_xreal_1,t127_real_2,t121_real_2,t33_euclid,t31_euclid,t33_euclid,t13_topreal3,d3_topreal1,d3_tarski,d12_funct_1,t13_topreal3,d6_weierstr,d12_funct_1,t13_topreal3,d6_weierstr,t5_metric_1,d2_seq_4,d1_funct_2,d6_weierstr,d12_funct_1,d5_seq_4,d3_weierstr,d5_weierstr,d8_weierstr,t45_jgraph_1,t13_jordan2c,t75_xcmplx_1,t88_xcmplx_1,t66_xreal_1,t50_xreal_1,t8_xreal_1,t70_xreal_1,l5_jordan2c,d1_absvalue,t10_xreal_1,t2_xreal_1,t8_toprns_1,t37_euclid,t34_euclid,t8_xreal_1,t1_xreal_1,t33_euclid,t33_euclid,t31_euclid,t26_toprns_1]), [file(jordan2c,t49_jordan2c),interesting(0.00)]). fof(t56_jordan2c,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [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_xreal_0(2,A) & C = a_2_2_jordan2c(A,B) & r2_hidden(D,C) & r2_hidden(E,C) & ~ ! [F] : ( m1_subset_1(F,k1_numbers) => ( D != k18_euclid(F,A,E) & E != k18_euclid(F,A,D) ) ) & ! [F] : ( m1_subset_1(F,u1_struct_0(k15_euclid(A))) => ! [G] : ( m1_subset_1(G,u1_struct_0(k15_euclid(A))) => ! [H] : ( m1_subset_1(H,u1_struct_0(k15_euclid(A))) => ! [I] : ( m1_subset_1(I,u1_struct_0(k15_euclid(A))) => ! [J] : ( m1_subset_1(J,u1_struct_0(k15_euclid(A))) => ~ ( r2_hidden(F,C) & r2_hidden(G,C) & r2_hidden(H,C) & r2_hidden(I,C) & r2_hidden(J,C) & r1_tarski(k3_topreal1(A,D,F),C) & r1_tarski(k3_topreal1(A,F,G),C) & r1_tarski(k3_topreal1(A,G,H),C) & r1_tarski(k3_topreal1(A,H,I),C) & r1_tarski(k3_topreal1(A,I,J),C) & r1_tarski(k3_topreal1(A,J,E),C) ) ) ) ) ) ) ) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t25_euclid,t31_xreal_1,t24_toprns_1,t55_jordan2c,t25_euclid,t12_euclid,t11_euclid,d11_euclid,t33_euclid,t127_real_2,d5_jgraph_1,t14_euclid,d1_absvalue,t88_xcmplx_1,t31_xreal_1,d11_euclid,t71_rvsum_1,t71_rvsum_1,d7_xcmplx_0,t74_rvsum_1,t49_jordan2c,t33_euclid,t24_toprns_1,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t34_euclid,t33_euclid,t24_toprns_1,d7_xcmplx_0,t33_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,d7_xcmplx_0,t33_euclid,t34_euclid,t49_jordan2c,t25_euclid,d3_tarski,t25_euclid,t26_toprns_1,d10_xboole_0]), [file(jordan2c,t56_jordan2c),interesting(0.00)]). fof(d6_connsp_3,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(k3_pre_topc(A,B)))) => k5_connsp_3(A,B,C) = C ) ) ) ), file(connsp_3,d6_connsp_3), [interesting(0.00)]). fof(t117_finseq_2,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m2_finseq_2(B,A,k4_finseq_2(1,A)) => ? [C] : ( m1_subset_1(C,A) & B = k12_finseq_1(A,C) ) ) ) ), file(finseq_2,t117_finseq_2), [interesting(0.00)]). fof(t109_finseq_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m2_finseq_2(C,B,k4_finseq_2(A,B)) => k3_finseq_1(C) = A ) ) ) ), file(finseq_2,t109_finseq_2), [interesting(0.00)]). fof(d1_jordan2b,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => ! [D] : ( m1_subset_1(D,k1_numbers) => ( D = k1_jordan2b(A,B,C) <=> ! [E] : ( m2_finseq_1(E,k1_numbers) => ( E = C => D = k4_finseq_4(k5_numbers,k1_numbers,E,B) ) ) ) ) ) ) ) ), file(jordan2b,d1_jordan2b), [interesting(0.00)]). fof(t79_rvsum_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_finseq_2(C,k1_numbers,k4_finseq_2(A,k1_numbers)) => k2_seq_1(k5_numbers,k1_numbers,k12_rvsum_1(A,C),B) = k7_square_1(k2_seq_1(k5_numbers,k1_numbers,C,B)) ) ) ) ), file(rvsum_1,t79_rvsum_1), [interesting(0.00)]). fof(t57_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( B = k9_finseq_1(A) <=> ( k3_finseq_1(B) = 1 & k1_funct_1(B,1) = A ) ) ) ), file(finseq_1,t57_finseq_1), [interesting(0.00)]). fof(t122_real_2,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ( ( ~ r1_xreal_0(0,A) & ~ r1_xreal_0(0,B) ) | ( ~ r1_xreal_0(A,0) & ~ r1_xreal_0(B,0) ) ) & r1_xreal_0(k3_xcmplx_0(A,B),0) ) ) ) ), file(real_2,t122_real_2), [interesting(0.00)]). fof(t73_finseq_2,theorem,( ! [A] : k2_finseq_2(1,A) = k9_finseq_1(A) ), file(finseq_2,t73_finseq_2), [interesting(0.00)]). fof(t69_rvsum_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => ! [B] : ( m1_subset_1(B,k1_numbers) => k9_rvsum_1(A,k12_finseq_1(k1_numbers,B)) = k12_finseq_1(k1_numbers,k11_binop_2(A,B)) ) ) ), file(rvsum_1,t69_rvsum_1), [interesting(0.00)]). fof(t67_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k14_euclid(1)))) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(1)))) => ( ( A = a_1_0_jordan2c(B) & C = A ) => ( v2_connsp_1(C,k15_euclid(1)) & ~ v6_tbsp_1(A,k14_euclid(1)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t65_jordan2c,d9_tbsp_1,t11_absvalue,t132_complex1,t8_xreal_1,t8_xreal_1,t122_real_2,t10_xreal_1,t2_xreal_1,t36_jordan2c,d11_euclid,t69_rvsum_1,t11_absvalue,t8_xreal_1,t2_xreal_1,t36_jordan2c,d11_euclid,t69_rvsum_1,t132_complex1,t8_xreal_1,t45_jgraph_1,t54_euclid,t8_toprns_1,t37_jordan2c,t11_absvalue,t83_square_1,t8_xreal_1,t2_xreal_1,t2_xreal_1]), [file(jordan2c,t67_jordan2c),interesting(0.00)]). fof(t68_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k14_euclid(1)))) => ! [B] : ( m1_subset_1(B,k1_numbers) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(1)))) => ( ( A = a_1_1_jordan2c(B) & C = A ) => ( v2_connsp_1(C,k15_euclid(1)) & ~ v6_tbsp_1(A,k14_euclid(1)) ) ) ) ) ) ), inference(mizar_proof,[status(thm)],[t66_jordan2c,d9_tbsp_1,t11_absvalue,t132_complex1,t8_xreal_1,t8_xreal_1,t122_real_2,t10_xreal_1,t2_xreal_1,t26_xreal_1,t36_jordan2c,d11_euclid,t69_rvsum_1,t11_absvalue,t8_xreal_1,t2_xreal_1,t26_xreal_1,t36_jordan2c,d11_euclid,t69_rvsum_1,t132_complex1,t8_xreal_1,t45_jgraph_1,t54_euclid,t27_toprns_1,t44_euclid,t8_toprns_1,t37_jordan2c,t11_absvalue,t83_square_1,t8_xreal_1,t2_xreal_1,t2_xreal_1]), [file(jordan2c,t68_jordan2c),interesting(0.00)]).