% Mizar ND problem: t89_jordan2c,jordan2c,3977,54 fof(dh_c1_96__jordan2c,definition, ( ( m2_subset_1(c1_96__jordan2c,k1_numbers,k5_numbers) => ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ) => ( v3_pre_topc(A,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),A)) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( ( ~ v1_xboole_0(C) & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(B)))) ) => ( v3_pre_topc(C,k15_euclid(B)) => v1_connsp_2(k3_pre_topc(k15_euclid(B),C)) ) ) ) ), introduced(definition,[new_symbol(c1_96__jordan2c),file(jordan2c,c1_96__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c1_96__jordan2c)]). fof(dh_c2_96__jordan2c,definition, ( ( ( ~ v1_xboole_0(c2_96__jordan2c) & m1_subset_1(c2_96__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ) => ( v3_pre_topc(c2_96__jordan2c,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) => ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ) => ( v3_pre_topc(A,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),A)) ) ) ), introduced(definition,[new_symbol(c2_96__jordan2c),file(jordan2c,c2_96__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c2_96__jordan2c)]). fof(e1_96__jordan2c,assumption,( v3_pre_topc(c2_96__jordan2c,k15_euclid(c1_96__jordan2c)) ), introduced(assumption,[file(jordan2c,e1_96__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,e1_96__jordan2c)]). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), [interesting(0.9),axiom,file(finseq_2,k4_finseq_2)]). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_u1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => ( v1_funct_1(u1_metric_1(A)) & v1_funct_2(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) & m2_relset_1(u1_metric_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) ) ) ), file(metric_1,u1_metric_1), [interesting(0.9),axiom,file(metric_1,u1_metric_1)]). fof(cc1_arytm_3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). fof(cc1_relset_1,theorem,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(cc2_arytm_3,theorem,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(fc14_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]). fof(fc4_subset_1,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]). fof(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). fof(rc1_jordan2c,theorem,( ! [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) ) ) ), file(jordan2c,rc1_jordan2c), [interesting(0.9),axiom,file(jordan2c,rc1_jordan2c)]). fof(rc2_finseq_1,theorem,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc2_finseq_1)]). fof(rc2_finset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), [interesting(0.9),axiom,file(finset_1,rc2_finset_1)]). fof(rc2_tbsp_1,theorem,( ! [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) ) ) ), file(tbsp_1,rc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,rc2_tbsp_1)]). fof(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_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(A))) & ~ v1_xboole_0(B) & v1_finset_1(B) & v6_tbsp_1(B,A) ) ) ), file(tbsp_1,rc3_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,rc3_tbsp_1)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(free_g1_metric_1,definition,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ! [C,D] : ( g1_metric_1(A,B) = g1_metric_1(C,D) => ( A = C & B = D ) ) ) ), file(metric_1,g1_metric_1), [interesting(0.9),axiom,file(metric_1,g1_metric_1)]). fof(free_g1_pre_topc,definition,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ! [C,D] : ( g1_pre_topc(A,B) = g1_pre_topc(C,D) => ( A = C & B = D ) ) ) ), file(pre_topc,g1_pre_topc), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(abstractness_v1_metric_1,theorem,( ! [A] : ( l1_metric_1(A) => ( v1_metric_1(A) => A = g1_metric_1(u1_struct_0(A),u1_metric_1(A)) ) ) ), file(metric_1,v1_metric_1), [interesting(0.9),axiom,file(metric_1,v1_metric_1)]). fof(existence_l1_metric_1,axiom,( ? [A] : l1_metric_1(A) ), file(metric_1,l1_metric_1), [interesting(0.9),axiom,file(metric_1,l1_metric_1)]). fof(dt_g1_metric_1,axiom,( ! [A,B] : ( ( v1_funct_1(B) & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers) & m1_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) ) => ( v1_metric_1(g1_metric_1(A,B)) & l1_metric_1(g1_metric_1(A,B)) ) ) ), file(metric_1,g1_metric_1), [interesting(0.9),axiom,file(metric_1,g1_metric_1)]). fof(dt_g1_pre_topc,axiom,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) => ( v1_pre_topc(g1_pre_topc(A,B)) & l1_pre_topc(g1_pre_topc(A,B)) ) ) ), file(pre_topc,g1_pre_topc), [interesting(0.9),axiom,file(pre_topc,g1_pre_topc)]). fof(dt_k13_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_funct_1(k13_euclid(A)) & v1_funct_2(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) & m2_relset_1(k13_euclid(A),k2_zfmisc_1(k1_euclid(A),k1_euclid(A)),k1_numbers) ) ) ), file(euclid,k13_euclid), [interesting(0.9),axiom,file(euclid,k13_euclid)]). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), [interesting(0.9),axiom,file(euclid,k1_euclid)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_l1_metric_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_struct_0(A) ) ), file(metric_1,l1_metric_1), [interesting(0.9),axiom,file(metric_1,l1_metric_1)]). fof(dt_u1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => m1_subset_1(u1_pre_topc(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ), file(pre_topc,u1_pre_topc), [interesting(0.9),axiom,file(pre_topc,u1_pre_topc)]). fof(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(cc2_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & v6_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(A))) => ( v1_finset_1(B) => v6_tbsp_1(B,A) ) ) ) ), file(tbsp_1,cc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,cc2_tbsp_1)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_arytm_3,theorem,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(fc3_pcomps_1,theorem,( ! [A] : ( l1_metric_1(A) => ( v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc3_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,fc3_pcomps_1)]). fof(fc4_pcomps_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( ~ v3_struct_0(k5_pcomps_1(A)) & v1_pre_topc(k5_pcomps_1(A)) & v2_pre_topc(k5_pcomps_1(A)) ) ) ), file(pcomps_1,fc4_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,fc4_pcomps_1)]). fof(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc1_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & v1_metric_1(A) ) ), file(metric_1,rc1_metric_1), [interesting(0.9),axiom,file(metric_1,rc1_metric_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(rc2_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) ) ), file(metric_1,rc2_metric_1), [interesting(0.9),axiom,file(metric_1,rc2_metric_1)]). fof(rc2_pcomps_1,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) & v3_compts_1(A) ) ), file(pcomps_1,rc2_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,rc2_pcomps_1)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(rc3_metric_1,theorem,( ? [A] : ( l1_metric_1(A) & ~ v3_struct_0(A) & v1_metric_1(A) & v6_metric_1(A) & v7_metric_1(A) & v8_metric_1(A) & v9_metric_1(A) ) ), file(metric_1,rc3_metric_1), [interesting(0.9),axiom,file(metric_1,rc3_metric_1)]). fof(rc4_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(rc8_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). fof(d1_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_euclid(A) = k4_finseq_2(A,k1_numbers) ) ), file(euclid,d1_euclid), [interesting(0.9),axiom,file(euclid,d1_euclid)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(abstractness_v1_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ( v1_pre_topc(A) => A = g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) ) ) ), file(pre_topc,v1_pre_topc), [interesting(0.9),axiom,file(pre_topc,v1_pre_topc)]). fof(existence_l1_struct_0,axiom,( ? [A] : l1_struct_0(A) ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(existence_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ? [B] : m1_pre_topc(B,A) ) ), file(pre_topc,m1_pre_topc), [interesting(0.9),axiom,file(pre_topc,m1_pre_topc)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k14_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) & l1_metric_1(k14_euclid(A)) ) ) ), file(euclid,k14_euclid), [interesting(0.9),axiom,file(euclid,k14_euclid)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_k5_pcomps_1,axiom,( ! [A] : ( l1_metric_1(A) => l1_pre_topc(k5_pcomps_1(A)) ) ), file(pcomps_1,k5_pcomps_1), [interesting(0.9),axiom,file(pcomps_1,k5_pcomps_1)]). fof(dt_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). fof(dt_m1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => ! [B] : ( m1_pre_topc(B,A) => l1_pre_topc(B) ) ) ), file(pre_topc,m1_pre_topc), [interesting(0.9),axiom,file(pre_topc,m1_pre_topc)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc1_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ! [B] : ( m1_pre_topc(B,A) => v2_pre_topc(B) ) ) ), file(pre_topc,cc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,cc1_pre_topc)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_finset_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc1_euclid,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k14_euclid(A)) & v1_metric_1(k14_euclid(A)) & v6_metric_1(k14_euclid(A)) & v7_metric_1(k14_euclid(A)) & v8_metric_1(k14_euclid(A)) & v9_metric_1(k14_euclid(A)) ) ) ), file(euclid,fc1_euclid), [interesting(0.9),axiom,file(euclid,fc1_euclid)]). fof(fc1_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(u1_struct_0(A)) ) ), file(struct_0,fc1_struct_0), [interesting(0.9),axiom,file(struct_0,fc1_struct_0)]). fof(fc2_euclid,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) ) ) ), file(euclid,fc2_euclid), [interesting(0.9),axiom,file(euclid,fc2_euclid)]). fof(fc2_finseq_1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(fc2_topreal1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v3_struct_0(k15_euclid(A)) & v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & v3_compts_1(k15_euclid(A)) ) ) ), file(topreal1,fc2_topreal1), [interesting(0.9),axiom,file(topreal1,fc2_topreal1)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & v1_pre_topc(A) ) ), file(pre_topc,rc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc1_pre_topc)]). fof(rc2_pre_topc,theorem,( ? [A] : ( l1_pre_topc(A) & ~ v3_struct_0(A) & v1_pre_topc(A) & v2_pre_topc(A) ) ), file(pre_topc,rc2_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc2_pre_topc)]). fof(rc3_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]). fof(rc3_pre_topc,theorem,( ! [A] : ( l1_pre_topc(A) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) ) ) ), file(pre_topc,rc3_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc3_pre_topc)]). fof(rc3_struct_0,theorem,( ? [A] : ( l1_struct_0(A) & ~ v3_struct_0(A) ) ), file(struct_0,rc3_struct_0), [interesting(0.9),axiom,file(struct_0,rc3_struct_0)]). fof(rc4_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), [interesting(0.9),axiom,file(finset_1,rc4_finset_1)]). fof(rc4_pre_topc,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & ~ v3_struct_0(B) & v1_pre_topc(B) ) ) ), file(pre_topc,rc4_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc4_pre_topc)]). fof(rc5_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_pre_topc(B,A) & v1_pre_topc(B) & v2_pre_topc(B) ) ) ), file(pre_topc,rc5_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc5_pre_topc)]). fof(rc5_struct_0,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & ~ v1_xboole_0(B) ) ) ), file(struct_0,rc5_struct_0), [interesting(0.9),axiom,file(struct_0,rc5_struct_0)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(d7_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k14_euclid(A) = g1_metric_1(k1_euclid(A),k13_euclid(A)) ) ), file(euclid,d7_euclid), [interesting(0.9),axiom,file(euclid,d7_euclid)]). fof(existence_l1_pre_topc,axiom,( ? [A] : l1_pre_topc(A) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_k15_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_pre_topc(k15_euclid(A)) & v2_pre_topc(k15_euclid(A)) & l1_pre_topc(k15_euclid(A)) ) ) ), file(euclid,k15_euclid), [interesting(0.9),axiom,file(euclid,k15_euclid)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k3_pre_topc,axiom,( ! [A,B] : ( ( l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & m1_pre_topc(k3_pre_topc(A,B),A) ) ) ), file(pre_topc,k3_pre_topc), [interesting(0.9),axiom,file(pre_topc,k3_pre_topc)]). fof(dt_l1_pre_topc,axiom,( ! [A] : ( l1_pre_topc(A) => l1_struct_0(A) ) ), file(pre_topc,l1_pre_topc), [interesting(0.9),axiom,file(pre_topc,l1_pre_topc)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). fof(dt_c1_96__jordan2c,assumption,( m2_subset_1(c1_96__jordan2c,k1_numbers,k5_numbers) ), introduced(assumption,[file(jordan2c,c1_96__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c1_96__jordan2c)]). fof(dt_c2_96__jordan2c,assumption, ( ~ v1_xboole_0(c2_96__jordan2c) & m1_subset_1(c2_96__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), introduced(assumption,[file(jordan2c,c2_96__jordan2c)]), [interesting(0.8),axiom,file(jordan2c,c2_96__jordan2c)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc1_finset_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]). fof(cc1_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]). fof(fc3_pre_topc,theorem,( ! [A,B] : ( ( ~ v3_struct_0(A) & l1_pre_topc(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( ~ v3_struct_0(k3_pre_topc(A,B)) & v1_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc3_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc3_pre_topc)]). fof(fc4_pre_topc,theorem,( ! [A,B] : ( ( v2_pre_topc(A) & l1_pre_topc(A) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) => ( v1_pre_topc(k3_pre_topc(A,B)) & v2_pre_topc(k3_pre_topc(A,B)) ) ) ), file(pre_topc,fc4_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc4_pre_topc)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). 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(0.9),axiom,file(euclid,d8_euclid)]). fof(dh_c1_96_1__jordan2c,definition, ( ( ( ~ v1_xboole_0(c1_96_1__jordan2c) & m1_subset_1(c1_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,A) ) => v3_pre_topc(A,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(B,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),B,C) ) => v3_pre_topc(C,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) ) ), introduced(definition,[new_symbol(c1_96_1__jordan2c),file(jordan2c,c1_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c1_96_1__jordan2c)]). fof(dh_c2_96_1__jordan2c,definition, ( ( m1_subset_1(c2_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,c2_96_1__jordan2c) ) => v3_pre_topc(c2_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,A) ) => v3_pre_topc(A,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) ), introduced(definition,[new_symbol(c2_96_1__jordan2c),file(jordan2c,c2_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c2_96_1__jordan2c)]). fof(e1_96_1__jordan2c,assumption, ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,c2_96_1__jordan2c) ), introduced(assumption,[file(jordan2c,e1_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,e1_96_1__jordan2c)]). fof(fc10_finset_1,theorem,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_1), [interesting(0.9),axiom,file(finset_1,fc10_finset_1)]). fof(fc11_finset_1,theorem,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), [interesting(0.9),axiom,file(finset_1,fc11_finset_1)]). fof(fc27_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), [interesting(0.9),axiom,file(membered,fc27_membered)]). fof(fc28_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), [interesting(0.9),axiom,file(membered,fc28_membered)]). fof(fc29_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), [interesting(0.9),axiom,file(membered,fc29_membered)]). fof(fc2_pre_topc,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) ) => ~ v1_xboole_0(k2_pre_topc(A)) ) ), file(pre_topc,fc2_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc2_pre_topc)]). fof(fc30_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), [interesting(0.9),axiom,file(membered,fc30_membered)]). fof(fc31_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), [interesting(0.9),axiom,file(membered,fc31_membered)]). fof(fc32_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), [interesting(0.9),axiom,file(membered,fc32_membered)]). fof(fc33_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), [interesting(0.9),axiom,file(membered,fc33_membered)]). fof(fc34_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), [interesting(0.9),axiom,file(membered,fc34_membered)]). fof(fc35_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), [interesting(0.9),axiom,file(membered,fc35_membered)]). fof(fc36_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), [interesting(0.9),axiom,file(membered,fc36_membered)]). fof(rc6_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc6_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc6_pre_topc)]). fof(rc7_pre_topc,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))) & ~ v1_xboole_0(B) & v4_pre_topc(B,A) ) ) ), file(pre_topc,rc7_pre_topc), [interesting(0.9),axiom,file(pre_topc,rc7_pre_topc)]). fof(t2_boole,theorem,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), [interesting(0.9),axiom,file(boole,t2_boole)]). fof(fc1_jordan2c,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k2_pre_topc(k15_euclid(A))) & v4_pre_topc(k2_pre_topc(k15_euclid(A)),k15_euclid(A)) & v2_connsp_1(k2_pre_topc(k15_euclid(A)),k15_euclid(A)) ) ) ), file(jordan2c,fc1_jordan2c), [interesting(0.9),axiom,file(jordan2c,fc1_jordan2c)]). fof(fc5_pre_topc,theorem,( ! [A] : ( ( v2_pre_topc(A) & l1_pre_topc(A) ) => v4_pre_topc(k2_pre_topc(A),A) ) ), file(pre_topc,fc5_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc5_pre_topc)]). fof(commutativity_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(idempotence_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(commutativity_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(idempotence_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(redefinition_k5_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(dt_k2_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k2_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k2_pre_topc), [interesting(0.9),axiom,file(pre_topc,k2_pre_topc)]). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(dt_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(dt_c2_96_1__jordan2c,assumption,( m1_subset_1(c2_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ), introduced(assumption,[file(jordan2c,c2_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c2_96_1__jordan2c)]). fof(de_c4_96_1__jordan2c,definition,( c4_96_1__jordan2c = c2_96_1__jordan2c ), introduced(definition,[new_symbol(c4_96_1__jordan2c),file(jordan2c,c4_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c4_96_1__jordan2c)]). fof(dt_c1_96_1__jordan2c,assumption, ( ~ v1_xboole_0(c1_96_1__jordan2c) & m1_subset_1(c1_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ), introduced(assumption,[file(jordan2c,c1_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c1_96_1__jordan2c)]). fof(dh_c3_96_1__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) & A = c2_96_1__jordan2c & r3_connsp_1(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c),A) ) => ( m1_subset_1(c3_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) & c3_96_1__jordan2c = c2_96_1__jordan2c & r3_connsp_1(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c),c3_96_1__jordan2c) ) ), introduced(definition,[new_symbol(c3_96_1__jordan2c),file(jordan2c,c3_96_1__jordan2c)]), [interesting(0.65),axiom,file(jordan2c,c3_96_1__jordan2c)]). 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(0.9),axiom,file(connsp_1,d6_connsp_1)]). fof(e2_96_1__jordan2c,plain,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) & A = c2_96_1__jordan2c & r3_connsp_1(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c),A) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,e1_96_1__jordan2c,d6_connsp_1]), [interesting(0.65),file(jordan2c,e2_96_1__jordan2c),[file(jordan2c,e2_96_1__jordan2c)]]). fof(dt_c3_96_1__jordan2c,plain,( m1_subset_1(c3_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) ), inference(consider,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dh_c3_96_1__jordan2c,e2_96_1__jordan2c]), [interesting(0.65),file(jordan2c,c3_96_1__jordan2c),[file(jordan2c,c3_96_1__jordan2c)]]). fof(e8_96_1__jordan2c,plain,( r1_tarski(c1_96_1__jordan2c,u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,dt_k15_euclid,dt_k3_pre_topc,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,t3_subset,d8_euclid]), [interesting(0.65),file(jordan2c,e8_96_1__jordan2c),[file(jordan2c,e8_96_1__jordan2c)]]). 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(0.9),axiom,file(jordan1,t1_jordan1)]). fof(e9_96_1__jordan2c,plain,( r1_tarski(c1_96_1__jordan2c,c2_96__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,fc1_subset_1,t3_subset,d8_euclid,e8_96_1__jordan2c,t1_jordan1]), [interesting(0.65),file(jordan2c,e9_96_1__jordan2c),[file(jordan2c,e9_96_1__jordan2c)]]). fof(e3_96_1__jordan2c,plain, ( c3_96_1__jordan2c = c2_96_1__jordan2c & r3_connsp_1(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c),c3_96_1__jordan2c) ), inference(consider,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dh_c3_96_1__jordan2c,e2_96_1__jordan2c]), [interesting(0.65),file(jordan2c,e3_96_1__jordan2c),[file(jordan2c,e3_96_1__jordan2c)]]). fof(e5_96_1__jordan2c,plain,( r1_tarski(c3_96_1__jordan2c,u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,dt_k15_euclid,dt_k3_pre_topc,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c3_96_1__jordan2c,t3_subset,d8_euclid]), [interesting(0.65),file(jordan2c,e5_96_1__jordan2c),[file(jordan2c,e5_96_1__jordan2c)]]). 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(0.9),axiom,file(pre_topc,t12_pre_topc)]). fof(e6_96_1__jordan2c,plain,( r1_tarski(c3_96_1__jordan2c,k2_pre_topc(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc2_pre_topc,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,fc5_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,existence_l1_struct_0,dt_k15_euclid,dt_k2_pre_topc,dt_k3_pre_topc,dt_l1_struct_0,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c3_96_1__jordan2c,t3_subset,d8_euclid,e5_96_1__jordan2c,t12_pre_topc]), [interesting(0.65),file(jordan2c,e6_96_1__jordan2c),[file(jordan2c,e6_96_1__jordan2c)]]). 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(0.9),axiom,file(pre_topc,d10_pre_topc)]). fof(e7_96_1__jordan2c,plain,( r1_tarski(c3_96_1__jordan2c,c1_96_1__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc2_pre_topc,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,d1_euclid,free_g1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_pre_topc,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m2_subset_1,dt_u1_pre_topc,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,fc5_pre_topc,rc1_subset_1,rc2_subset_1,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c3_96_1__jordan2c,fc1_subset_1,rc1_pre_topc,rc3_pre_topc,t3_subset,d8_euclid,e6_96_1__jordan2c,d10_pre_topc]), [interesting(0.65),file(jordan2c,e7_96_1__jordan2c),[file(jordan2c,e7_96_1__jordan2c)]]). 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(0.9),axiom,file(xboole_1,t1_xboole_1)]). fof(e10_96_1__jordan2c,plain,( r1_tarski(c2_96_1__jordan2c,c2_96__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_l1_struct_0,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,dt_k15_euclid,dt_k3_pre_topc,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,dt_c3_96_1__jordan2c,t3_subset,d8_euclid,e9_96_1__jordan2c,e3_96_1__jordan2c,e7_96_1__jordan2c,t1_xboole_1]), [interesting(0.65),file(jordan2c,e10_96_1__jordan2c),[file(jordan2c,e10_96_1__jordan2c)]]). fof(e11_96_1__jordan2c,plain,( m1_subset_1(c2_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_pre_topc,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_m1_pre_topc,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_pre_topc,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k3_pre_topc,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,e10_96_1__jordan2c,t1_xboole_1]), [interesting(0.65),file(jordan2c,e11_96_1__jordan2c),[file(jordan2c,e11_96_1__jordan2c)]]). fof(dt_c4_96_1__jordan2c,plain,( m1_subset_1(c4_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_pre_topc,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_m1_pre_topc,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_pre_topc,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k3_pre_topc,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,dt_c2_96__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,de_c4_96_1__jordan2c,e11_96_1__jordan2c]), [interesting(0.65),file(jordan2c,c4_96_1__jordan2c),[file(jordan2c,c4_96_1__jordan2c)]]). fof(e14_96_1__jordan2c,plain,( r1_tarski(c2_96_1__jordan2c,u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,dt_k15_euclid,dt_k3_pre_topc,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,t3_subset,d8_euclid]), [interesting(0.65),file(jordan2c,e14_96_1__jordan2c),[file(jordan2c,e14_96_1__jordan2c)]]). fof(e15_96_1__jordan2c,plain,( r1_tarski(c2_96_1__jordan2c,c2_96__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,e14_96_1__jordan2c,t1_jordan1]), [interesting(0.65),file(jordan2c,e15_96_1__jordan2c),[file(jordan2c,e15_96_1__jordan2c)]]). fof(t28_xboole_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => k3_xboole_0(A,B) = A ) ), file(xboole_1,t28_xboole_1), [interesting(0.9),axiom,file(xboole_1,t28_xboole_1)]). fof(e16_96_1__jordan2c,plain,( k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c4_96_1__jordan2c,c2_96__jordan2c) = c2_96_1__jordan2c ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1__jordan2c,e1_96_1__jordan2c,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_pre_topc,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_m1_pre_topc,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc10_finset_1,fc11_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_pre_topc,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t1_subset,t2_boole,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,reflexivity_r1_tarski,redefinition_k5_subset_1,dt_k15_euclid,dt_k3_xboole_0,dt_k5_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,dt_c4_96_1__jordan2c,de_c4_96_1__jordan2c,t3_subset,d8_euclid,e15_96_1__jordan2c,t28_xboole_1]), [interesting(0.65),file(jordan2c,e16_96_1__jordan2c),[file(jordan2c,e16_96_1__jordan2c)]]). fof(e4_96_1__jordan2c,plain,( k2_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) = c2_96__jordan2c ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c2_96__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc2_pre_topc,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,d1_euclid,free_g1_pre_topc,reflexivity_r1_tarski,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_pre_topc,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m2_subset_1,dt_u1_pre_topc,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,fc5_pre_topc,rc1_subset_1,rc2_subset_1,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,fc1_subset_1,rc1_pre_topc,rc3_pre_topc,t3_subset,d8_euclid,d10_pre_topc]), [interesting(0.65),file(jordan2c,e4_96_1__jordan2c),[file(jordan2c,e4_96_1__jordan2c)]]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(dt_k9_metric_1,axiom,( ! [A,B,C] : ( ( l1_metric_1(A) & m1_subset_1(B,u1_struct_0(A)) & v1_xreal_0(C) ) => m1_subset_1(k9_metric_1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) ), file(metric_1,k9_metric_1), [interesting(0.9),axiom,file(metric_1,k9_metric_1)]). fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(dh_c1_96_1_1__jordan2c,definition, ( ( m1_subset_1(c1_96_1_1__jordan2c,u1_struct_0(k14_euclid(c1_96__jordan2c))) => ~ ( r2_hidden(c1_96_1_1__jordan2c,c4_96_1__jordan2c) & ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),c4_96_1__jordan2c) ) ) ) ) => ! [B] : ( m1_subset_1(B,u1_struct_0(k14_euclid(c1_96__jordan2c))) => ~ ( r2_hidden(B,c4_96_1__jordan2c) & ! [C] : ( v1_xreal_0(C) => ~ ( ~ r1_xreal_0(C,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),B,C),c4_96_1__jordan2c) ) ) ) ) ), introduced(definition,[new_symbol(c1_96_1_1__jordan2c),file(jordan2c,c1_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c1_96_1_1__jordan2c)]). fof(e1_96_1_1__jordan2c,assumption,( r2_hidden(c1_96_1_1__jordan2c,c4_96_1__jordan2c) ), introduced(assumption,[file(jordan2c,e1_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,e1_96_1_1__jordan2c)]). fof(dt_c1_96_1_1__jordan2c,assumption,( m1_subset_1(c1_96_1_1__jordan2c,u1_struct_0(k14_euclid(c1_96__jordan2c))) ), introduced(assumption,[file(jordan2c,c1_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c1_96_1_1__jordan2c)]). fof(dh_c2_96_1_1__jordan2c,definition, ( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) & v3_pre_topc(A,k15_euclid(c1_96__jordan2c)) & k3_xboole_0(A,k2_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c))) = c1_96_1__jordan2c ) => ( m1_subset_1(c2_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) & v3_pre_topc(c2_96_1_1__jordan2c,k15_euclid(c1_96__jordan2c)) & k3_xboole_0(c2_96_1_1__jordan2c,k2_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c))) = c1_96_1__jordan2c ) ), introduced(definition,[new_symbol(c2_96_1_1__jordan2c),file(jordan2c,c2_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c2_96_1_1__jordan2c)]). 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(0.9),axiom,file(tops_2,t32_tops_2)]). fof(e2_96_1_1__jordan2c,plain,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) & v3_pre_topc(A,k15_euclid(c1_96__jordan2c)) & k3_xboole_0(A,k2_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c))) = c1_96_1__jordan2c ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc10_finset_1,fc11_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_boole,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,fc5_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k3_xboole_0,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,e1_96_1__jordan2c,t32_tops_2]), [interesting(0.5),file(jordan2c,e2_96_1_1__jordan2c),[file(jordan2c,e2_96_1_1__jordan2c)]]). fof(dt_c2_96_1_1__jordan2c,plain,( m1_subset_1(c2_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(consider,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dh_c2_96_1_1__jordan2c,e2_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c2_96_1_1__jordan2c),[file(jordan2c,c2_96_1_1__jordan2c)]]). fof(dh_c3_96_1_1__jordan2c,definition, ( ? [A] : ( v1_xreal_0(A) & ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c2_96_1_1__jordan2c,c2_96__jordan2c)) ) => ( v1_xreal_0(c3_96_1_1__jordan2c) & ~ r1_xreal_0(c3_96_1_1__jordan2c,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c3_96_1_1__jordan2c),k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c2_96_1_1__jordan2c,c2_96__jordan2c)) ) ), introduced(definition,[new_symbol(c3_96_1_1__jordan2c),file(jordan2c,c3_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c3_96_1_1__jordan2c)]). fof(e3_96_1_1__jordan2c,plain, ( v3_pre_topc(c2_96_1_1__jordan2c,k15_euclid(c1_96__jordan2c)) & k3_xboole_0(c2_96_1_1__jordan2c,k2_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c))) = c1_96_1__jordan2c ), inference(consider,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dh_c2_96_1_1__jordan2c,e2_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,e3_96_1_1__jordan2c),[file(jordan2c,e3_96_1_1__jordan2c)]]). 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(0.9),axiom,file(tops_1,t38_tops_1)]). fof(e5_96_1_1__jordan2c,plain,( v3_pre_topc(k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c2_96_1_1__jordan2c,c2_96__jordan2c),k15_euclid(c1_96__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc10_finset_1,fc11_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc7_pre_topc,t1_subset,t2_boole,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,rc6_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,existence_l1_pre_topc,existence_m1_subset_1,redefinition_k5_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1_1__jordan2c,fc1_subset_1,fc4_pre_topc,fc5_pre_topc,t3_subset,d8_euclid,e1_96__jordan2c,e3_96_1_1__jordan2c,t38_tops_1]), [interesting(0.5),file(jordan2c,e5_96_1_1__jordan2c),[file(jordan2c,e5_96_1_1__jordan2c)]]). fof(e4_96_1_1__jordan2c,plain,( k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c2_96_1_1__jordan2c,c2_96__jordan2c) = c1_96_1__jordan2c ), inference(mizar_by,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc10_finset_1,fc11_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_boole,t4_subset,t5_subset,d1_euclid,free_g1_pre_topc,reflexivity_r1_tarski,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_pre_topc,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m2_subset_1,dt_u1_pre_topc,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,fc5_pre_topc,rc1_subset_1,rc2_subset_1,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,redefinition_k5_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1_1__jordan2c,fc1_subset_1,rc1_pre_topc,rc3_pre_topc,t3_subset,d8_euclid,e3_96_1_1__jordan2c,d10_pre_topc]), [interesting(0.5),file(jordan2c,e4_96_1_1__jordan2c),[file(jordan2c,e4_96_1_1__jordan2c)]]). 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(0.9),axiom,file(topmetr,t22_topmetr)]). fof(e6_96_1_1__jordan2c,plain,( ? [A] : ( v1_xreal_0(A) & ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c2_96_1_1__jordan2c,c2_96__jordan2c)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_pcomps_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_boole,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,free_g1_metric_1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k3_xboole_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pre_topc,fc4_pre_topc,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_metric_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t1_numerals,t1_real,t2_subset,t4_real,t5_subset,t6_boole,t8_boole,d1_euclid,commutativity_k5_subset_1,idempotence_k5_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_l1_metric_1,existence_m1_subset_1,redefinition_k5_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k5_pcomps_1,dt_k5_subset_1,dt_k9_metric_1,dt_l1_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,dt_c2_96_1_1__jordan2c,dt_c3_96_1__jordan2c,dt_c4_96_1__jordan2c,de_c4_96_1__jordan2c,cc2_xreal_0,fc1_subset_1,fc3_pcomps_1,fc4_pcomps_1,rqLessOrEqual__r1_xreal_0__r0_r0,t1_subset,t3_subset,t4_subset,t7_boole,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e5_96_1_1__jordan2c,e3_96_1__jordan2c,e7_96_1__jordan2c,e1_96_1_1__jordan2c,e4_96_1_1__jordan2c,t22_topmetr]), [interesting(0.5),file(jordan2c,e6_96_1_1__jordan2c),[file(jordan2c,e6_96_1_1__jordan2c)]]). fof(dt_c3_96_1_1__jordan2c,plain,( v1_xreal_0(c3_96_1_1__jordan2c) ), inference(consider,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dh_c3_96_1_1__jordan2c,e6_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c3_96_1_1__jordan2c),[file(jordan2c,c3_96_1_1__jordan2c)]]). fof(de_c4_96_1_1__jordan2c,definition,( c4_96_1_1__jordan2c = c3_96_1_1__jordan2c ), introduced(definition,[new_symbol(c4_96_1_1__jordan2c),file(jordan2c,c4_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c4_96_1_1__jordan2c)]). fof(d1_xreal_0,definition,( ! [A] : ( v1_xreal_0(A) <=> r2_hidden(A,k1_numbers) ) ), file(xreal_0,d1_xreal_0), [interesting(0.9),axiom,file(xreal_0,d1_xreal_0)]). fof(e8_96_1_1__jordan2c,plain,( m1_subset_1(c3_96_1_1__jordan2c,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_finseq_1,cc1_xreal_0,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t8_boole,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_funct_1,cc4_membered,cc7_xreal_0,rc1_xreal_0,t2_subset,t6_boole,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_numbers,dt_m1_subset_1,dt_c3_96_1_1__jordan2c,cc2_xreal_0,fc2_membered,t1_subset,t7_boole,d1_xreal_0]), [interesting(0.5),file(jordan2c,e8_96_1_1__jordan2c),[file(jordan2c,e8_96_1_1__jordan2c)]]). fof(dt_c4_96_1_1__jordan2c,plain,( m1_subset_1(c4_96_1_1__jordan2c,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_finseq_1,cc1_xreal_0,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc4_membered,cc7_xreal_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,dt_k1_numbers,dt_m1_subset_1,dt_c3_96_1_1__jordan2c,fc2_membered,de_c4_96_1_1__jordan2c,e8_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c4_96_1_1__jordan2c),[file(jordan2c,c4_96_1_1__jordan2c)]]). fof(e1_96_1_1_1__jordan2c,assumption,( ~ r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),c4_96_1__jordan2c) ), introduced(assumption,[file(jordan2c,e1_96_1_1_1__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,e1_96_1_1_1__jordan2c)]). fof(de_c9_96_1_1__jordan2c,definition,( c9_96_1_1__jordan2c = k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c) ), introduced(definition,[new_symbol(c9_96_1_1__jordan2c),file(jordan2c,c9_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c9_96_1_1__jordan2c)]). fof(e7_96_1_1__jordan2c,plain, ( ~ r1_xreal_0(c3_96_1_1__jordan2c,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c3_96_1_1__jordan2c),k5_subset_1(u1_struct_0(k15_euclid(c1_96__jordan2c)),c2_96_1_1__jordan2c,c2_96__jordan2c)) ), inference(consider,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dh_c3_96_1_1__jordan2c,e6_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,e7_96_1_1__jordan2c),[file(jordan2c,e7_96_1_1__jordan2c)]]). fof(e9_96_1_1__jordan2c,plain, ( ~ r1_xreal_0(c4_96_1_1__jordan2c,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),c1_96_1__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pre_topc,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_boole,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,dt_u1_pre_topc,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc4_pre_topc,fc5_pre_topc,rc1_metric_1,rc1_subset_1,rc2_subset_1,rc5_pre_topc,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,d1_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,redefinition_k5_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_k9_metric_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1_1__jordan2c,dt_c3_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,rc1_pre_topc,rc3_pre_topc,rqLessOrEqual__r1_xreal_0__r0_r0,t3_subset,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e3_96_1_1__jordan2c,e7_96_1_1__jordan2c,d10_pre_topc]), [interesting(0.5),file(jordan2c,e9_96_1_1__jordan2c),[file(jordan2c,e9_96_1_1__jordan2c)]]). fof(e14_96_1_1__jordan2c,plain,( r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),k2_pre_topc(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc2_pre_topc,fc3_pre_topc,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_struct_0,dt_m2_subset_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc4_pre_topc,fc5_pre_topc,rc1_metric_1,rc1_subset_1,rc2_subset_1,rc5_pre_topc,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,d1_euclid,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k9_metric_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,rc1_pre_topc,rc3_pre_topc,rqLessOrEqual__r1_xreal_0__r0_r0,t3_subset,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e9_96_1_1__jordan2c,d10_pre_topc]), [interesting(0.5),file(jordan2c,e14_96_1_1__jordan2c),[file(jordan2c,e14_96_1_1__jordan2c)]]). fof(e15_96_1_1__jordan2c,plain,( m1_subset_1(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc2_pre_topc,fc3_pre_topc,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t4_subset,t5_subset,free_g1_metric_1,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m2_subset_1,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc2_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc4_pre_topc,fc5_pre_topc,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d1_euclid,reflexivity_r1_tarski,existence_l1_struct_0,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k9_metric_1,dt_l1_struct_0,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,t3_subset,d7_euclid,d8_euclid,e14_96_1_1__jordan2c,t12_pre_topc]), [interesting(0.5),file(jordan2c,e15_96_1_1__jordan2c),[file(jordan2c,e15_96_1_1__jordan2c)]]). fof(dt_c9_96_1_1__jordan2c,plain,( m1_subset_1(c9_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc3_pre_topc,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc2_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc4_pre_topc,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d1_euclid,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k9_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,t3_subset,d7_euclid,d8_euclid,de_c9_96_1_1__jordan2c,e15_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c9_96_1_1__jordan2c),[file(jordan2c,c9_96_1_1__jordan2c)]]). fof(de_c10_96_1_1__jordan2c,definition,( c10_96_1_1__jordan2c = c9_96_1_1__jordan2c ), introduced(definition,[new_symbol(c10_96_1_1__jordan2c),file(jordan2c,c10_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c10_96_1_1__jordan2c)]). fof(e16_96_1_1__jordan2c,plain,( m1_subset_1(c9_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid]), [interesting(0.5),file(jordan2c,e16_96_1_1__jordan2c),[file(jordan2c,e16_96_1_1__jordan2c)]]). fof(dt_c10_96_1_1__jordan2c,plain,( m1_subset_1(c10_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,de_c10_96_1_1__jordan2c,e16_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c10_96_1_1__jordan2c),[file(jordan2c,c10_96_1_1__jordan2c)]]). fof(dh_c1_96_1_1_1__jordan2c,definition, ( ? [A] : ( r2_hidden(A,k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c)) & ~ r2_hidden(A,c4_96_1__jordan2c) ) => ( r2_hidden(c1_96_1_1_1__jordan2c,k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c)) & ~ r2_hidden(c1_96_1_1_1__jordan2c,c4_96_1__jordan2c) ) ), introduced(definition,[new_symbol(c1_96_1_1_1__jordan2c),file(jordan2c,c1_96_1_1_1__jordan2c)]), [interesting(0.35),axiom,file(jordan2c,c1_96_1_1_1__jordan2c)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(e2_96_1_1_1__jordan2c,plain,( ? [A] : ( r2_hidden(A,k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c)) & ~ r2_hidden(A,c4_96_1__jordan2c) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1_1__jordan2c])],[free_g1_pre_topc,existence_m1_pre_topc,dt_g1_pre_topc,dt_m1_pre_topc,dt_u1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_pre_topc,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_pre_topc,rc3_tbsp_1,rc4_funct_1,rc4_pre_topc,rc5_pre_topc,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k3_pre_topc,dt_k4_finseq_2,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c2_96__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc4_pre_topc,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_pre_topc,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,free_g1_metric_1,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_l1_metric_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c2_96_1__jordan2c,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_metric_1,rc1_subset_1,rc2_subset_1,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d1_euclid,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k14_euclid,dt_k9_metric_1,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c4_96_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1__jordan2c,de_c4_96_1_1__jordan2c,t1_subset,t3_subset,t7_boole,d7_euclid,e1_96_1_1_1__jordan2c,d3_tarski]), [interesting(0.35),file(jordan2c,e2_96_1_1_1__jordan2c),[file(jordan2c,e2_96_1_1_1__jordan2c)]]). fof(dt_c1_96_1_1_1__jordan2c,plain,( $true ), inference(consider,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1_1__jordan2c])],[dh_c1_96_1_1_1__jordan2c,e2_96_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,c1_96_1_1_1__jordan2c),[file(jordan2c,c1_96_1_1_1__jordan2c)]]). fof(symmetry_r1_xboole_0,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), [interesting(0.9),axiom,file(xboole_0,r1_xboole_0)]). fof(fc2_tbsp_1,theorem,( ! [A] : ( ( ~ v3_struct_0(A) & l1_metric_1(A) ) => ( v1_xboole_0(k1_pre_topc(A)) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) & v1_finset_1(k1_pre_topc(A)) & v6_tbsp_1(k1_pre_topc(A),A) ) ) ), file(tbsp_1,fc2_tbsp_1), [interesting(0.9),axiom,file(tbsp_1,fc2_tbsp_1)]). fof(fc1_pre_topc,theorem,( ! [A] : ( l1_struct_0(A) => ( v1_xboole_0(k1_pre_topc(A)) & v1_membered(k1_pre_topc(A)) & v2_membered(k1_pre_topc(A)) & v3_membered(k1_pre_topc(A)) & v4_membered(k1_pre_topc(A)) & v5_membered(k1_pre_topc(A)) ) ) ), file(pre_topc,fc1_pre_topc), [interesting(0.9),axiom,file(pre_topc,fc1_pre_topc)]). fof(dt_k1_pre_topc,axiom,( ! [A] : ( l1_struct_0(A) => m1_subset_1(k1_pre_topc(A),k1_zfmisc_1(u1_struct_0(A))) ) ), file(pre_topc,k1_pre_topc), [interesting(0.9),axiom,file(pre_topc,k1_pre_topc)]). 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(0.9),axiom,file(goboard6,t4_goboard6)]). fof(e8_96_1_1_1__jordan2c,plain,( r2_hidden(c1_96_1_1__jordan2c,c10_96_1_1__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,existence_m1_finseq_2,existence_m1_pre_topc,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_pre_topc,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_pcomps_1,rc2_xreal_0,rc3_finset_1,rc3_pre_topc,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_pre_topc,t2_boole,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,free_g1_metric_1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k3_xboole_0,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,dt_c1_96_1__jordan2c,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc4_pre_topc,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_metric_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t1_numerals,t1_real,t2_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d1_euclid,commutativity_k5_subset_1,idempotence_k5_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_l1_metric_1,existence_m1_subset_1,redefinition_k5_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k5_subset_1,dt_k9_metric_1,dt_l1_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c10_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1_1__jordan2c,dt_c3_96_1_1__jordan2c,de_c10_96_1_1__jordan2c,cc2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,t1_subset,t3_subset,t7_boole,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e7_96_1_1__jordan2c,t4_goboard6]), [interesting(0.35),file(jordan2c,e8_96_1_1_1__jordan2c),[file(jordan2c,e8_96_1_1_1__jordan2c)]]). 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(0.9),axiom,file(xboole_0,d3_xboole_0)]). fof(e9_96_1_1_1__jordan2c,plain,( k3_xboole_0(c10_96_1_1__jordan2c,c2_96_1__jordan2c) != k1_pre_topc(k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c3_96_1_1__jordan2c,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k9_metric_1,dt_l1_metric_1,dt_u1_pre_topc,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc10_finset_1,fc11_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_tbsp_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t2_boole,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_pre_topc,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k15_euclid,dt_k1_pre_topc,dt_k3_pre_topc,dt_k3_xboole_0,dt_c10_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,dt_c4_96_1__jordan2c,de_c10_96_1_1__jordan2c,de_c4_96_1__jordan2c,t1_subset,t7_boole,d8_euclid,e8_96_1_1_1__jordan2c,e1_96_1_1__jordan2c,d3_xboole_0]), [interesting(0.35),file(jordan2c,e9_96_1_1_1__jordan2c),[file(jordan2c,e9_96_1_1_1__jordan2c)]]). 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(0.9),axiom,file(xboole_0,d7_xboole_0)]). fof(e10_96_1_1_1__jordan2c,plain,( ~ r1_xboole_0(c10_96_1_1__jordan2c,c2_96_1__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c3_96_1_1__jordan2c,cc1_arytm_3,cc1_relset_1,cc2_arytm_3,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_tbsp_1,rc2_xreal_0,rc3_tbsp_1,rc3_xreal_0,rc4_xreal_0,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k5_ordinal2,dt_k9_metric_1,dt_l1_metric_1,dt_u1_pre_topc,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc1_xreal_0,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc7_xreal_0,fc1_struct_0,fc2_tbsp_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_metric_1,rc3_struct_0,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_pre_topc,cc20_membered,cc2_finset_1,cc2_funct_1,cc2_membered,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc10_finset_1,fc11_finset_1,fc1_euclid,fc1_pre_topc,fc1_subset_1,fc27_membered,fc28_membered,fc29_membered,fc2_euclid,fc2_membered,fc2_topreal1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc4_pre_topc,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_pre_topc,rc1_subset_1,rc2_funct_1,rc2_subset_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_pre_topc,rc4_finset_1,rc4_funct_1,rc5_pre_topc,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t7_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,symmetry_r1_xboole_0,dt_k15_euclid,dt_k1_pre_topc,dt_k1_xboole_0,dt_k3_pre_topc,dt_k3_xboole_0,dt_c10_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,de_c10_96_1_1__jordan2c,fc2_finseq_1,fc6_membered,t2_boole,t6_boole,d8_euclid,e9_96_1_1_1__jordan2c,d7_xboole_0]), [interesting(0.35),file(jordan2c,e10_96_1_1_1__jordan2c),[file(jordan2c,e10_96_1_1_1__jordan2c)]]). fof(de_c7_96_1_1__jordan2c,definition,( c7_96_1_1__jordan2c = k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c) ), introduced(definition,[new_symbol(c7_96_1_1__jordan2c),file(jordan2c,c7_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c7_96_1_1__jordan2c)]). fof(e12_96_1_1__jordan2c,plain,( m1_subset_1(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc3_pre_topc,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,free_g1_metric_1,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc4_pre_topc,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,d1_euclid,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k9_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,rqLessOrEqual__r1_xreal_0__r0_r0,t3_subset,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e9_96_1_1__jordan2c,t1_xboole_1]), [interesting(0.5),file(jordan2c,e12_96_1_1__jordan2c),[file(jordan2c,e12_96_1_1__jordan2c)]]). fof(dt_c7_96_1_1__jordan2c,plain,( m1_subset_1(c7_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc3_pre_topc,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc2_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,fc4_pre_topc,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d1_euclid,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k9_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,t3_subset,d7_euclid,d8_euclid,de_c7_96_1_1__jordan2c,e12_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c7_96_1_1__jordan2c),[file(jordan2c,c7_96_1_1__jordan2c)]]). fof(de_c8_96_1_1__jordan2c,definition,( c8_96_1_1__jordan2c = c7_96_1_1__jordan2c ), introduced(definition,[new_symbol(c8_96_1_1__jordan2c),file(jordan2c,c8_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c8_96_1_1__jordan2c)]). fof(e13_96_1_1__jordan2c,plain,( m1_subset_1(c7_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c7_96_1_1__jordan2c,de_c7_96_1_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid]), [interesting(0.5),file(jordan2c,e13_96_1_1__jordan2c),[file(jordan2c,e13_96_1_1__jordan2c)]]). fof(dt_c8_96_1_1__jordan2c,plain,( m1_subset_1(c8_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c7_96_1_1__jordan2c,de_c7_96_1_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,de_c8_96_1_1__jordan2c,e13_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c8_96_1_1__jordan2c),[file(jordan2c,c8_96_1_1__jordan2c)]]). fof(de_c5_96_1_1__jordan2c,definition,( c5_96_1_1__jordan2c = k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c) ), introduced(definition,[new_symbol(c5_96_1_1__jordan2c),file(jordan2c,c5_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c5_96_1_1__jordan2c)]). 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(0.9),axiom,file(topreal3,t13_topreal3)]). fof(e10_96_1_1__jordan2c,plain,( m1_subset_1(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc1_arytm_3,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc2_finseq_1,fc4_subset_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t4_subset,t5_subset,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,rc1_membered,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,d1_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k9_metric_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,t3_subset,d7_euclid,d8_euclid,t13_topreal3]), [interesting(0.5),file(jordan2c,e10_96_1_1__jordan2c),[file(jordan2c,e10_96_1_1__jordan2c)]]). fof(dt_c5_96_1_1__jordan2c,plain,( m1_subset_1(c5_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc4_pcomps_1,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,dt_c3_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,d1_euclid,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k9_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,fc1_subset_1,t3_subset,d7_euclid,d8_euclid,de_c5_96_1_1__jordan2c,e10_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c5_96_1_1__jordan2c),[file(jordan2c,c5_96_1_1__jordan2c)]]). fof(de_c6_96_1_1__jordan2c,definition,( c6_96_1_1__jordan2c = c5_96_1_1__jordan2c ), introduced(definition,[new_symbol(c6_96_1_1__jordan2c),file(jordan2c,c6_96_1_1__jordan2c)]), [interesting(0.5),axiom,file(jordan2c,c6_96_1_1__jordan2c)]). fof(e11_96_1_1__jordan2c,plain,( m1_subset_1(c5_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c5_96_1_1__jordan2c,de_c5_96_1_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid]), [interesting(0.5),file(jordan2c,e11_96_1_1__jordan2c),[file(jordan2c,e11_96_1_1__jordan2c)]]). fof(dt_c6_96_1_1__jordan2c,plain,( m1_subset_1(c6_96_1_1__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,dt_c3_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c5_96_1_1__jordan2c,de_c5_96_1_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,de_c6_96_1_1__jordan2c,e11_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,c6_96_1_1__jordan2c),[file(jordan2c,c6_96_1_1__jordan2c)]]). 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) ) ) ) ) ) ), file(jordan2c,t78_jordan2c), [interesting(0.9),axiom,file(jordan2c,t78_jordan2c)]). fof(e4_96_1_1_1__jordan2c,plain,( v1_jordan1(c6_96_1_1__jordan2c,c1_96__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[dt_c3_96_1_1__jordan2c,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc1_arytm_3,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc2_finseq_1,fc4_subset_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_xreal_0,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t4_subset,t5_subset,free_g1_metric_1,reflexivity_r1_tarski,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_c5_96_1_1__jordan2c,de_c5_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc4_membered,cc7_xreal_0,fc1_struct_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,rc1_membered,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,d1_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_k9_metric_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c6_96_1_1__jordan2c,de_c6_96_1_1__jordan2c,cc6_membered,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,t3_subset,d7_euclid,d8_euclid,t78_jordan2c]), [interesting(0.35),file(jordan2c,e4_96_1_1_1__jordan2c),[file(jordan2c,e4_96_1_1_1__jordan2c)]]). 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)) ) ) ) ), file(jordan2c,t14_jordan2c), [interesting(0.9),axiom,file(jordan2c,t14_jordan2c)]). fof(e5_96_1_1_1__jordan2c,plain,( v2_connsp_1(c6_96_1_1__jordan2c,k15_euclid(c1_96__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c3_96_1_1__jordan2c,cc1_finseq_1,cc1_relset_1,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k9_metric_1,dt_l1_metric_1,dt_u1_pre_topc,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc2_finset_1,cc2_tbsp_1,cc2_xreal_0,cc7_xreal_0,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_arytm_3,rc1_finset_1,rc1_metric_1,rc1_xreal_0,rc2_metric_1,rc3_finset_1,rc3_metric_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_c5_96_1_1__jordan2c,de_c5_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,fc1_euclid,fc1_struct_0,fc5_membered,rc1_membered,rc1_pre_topc,rc1_subset_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc3_struct_0,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c6_96_1_1__jordan2c,de_c6_96_1_1__jordan2c,cc6_membered,cc9_membered,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,t3_subset,d8_euclid,e4_96_1_1_1__jordan2c,t14_jordan2c]), [interesting(0.35),file(jordan2c,e5_96_1_1_1__jordan2c),[file(jordan2c,e5_96_1_1_1__jordan2c)]]). 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)) ) ) ) ) ) ), file(jordan2c,t15_jordan2c), [interesting(0.9),axiom,file(jordan2c,t15_jordan2c)]). fof(e6_96_1_1_1__jordan2c,plain,( v2_connsp_1(c8_96_1_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c3_96_1_1__jordan2c,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k9_metric_1,dt_l1_metric_1,dt_u1_pre_topc,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc3_finset_1,rc3_metric_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c5_96_1_1__jordan2c,dt_c7_96_1_1__jordan2c,de_c5_96_1_1__jordan2c,de_c7_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_struct_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_pre_topc,rc2_subset_1,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c6_96_1_1__jordan2c,dt_c8_96_1_1__jordan2c,de_c6_96_1_1__jordan2c,de_c8_96_1_1__jordan2c,fc1_subset_1,fc4_pre_topc,t3_subset,d8_euclid,e5_96_1_1_1__jordan2c,t15_jordan2c]), [interesting(0.35),file(jordan2c,e6_96_1_1_1__jordan2c),[file(jordan2c,e6_96_1_1_1__jordan2c)]]). fof(e7_96_1_1_1__jordan2c,plain,( v2_connsp_1(c10_96_1_1__jordan2c,k3_pre_topc(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c3_96_1_1__jordan2c,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k9_metric_1,dt_l1_metric_1,dt_u1_pre_topc,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc2_finseq_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc3_finset_1,rc3_metric_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c7_96_1_1__jordan2c,dt_c9_96_1_1__jordan2c,de_c7_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_struct_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_pre_topc,rc2_subset_1,rc3_pre_topc,rc3_struct_0,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c10_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c8_96_1_1__jordan2c,de_c10_96_1_1__jordan2c,de_c8_96_1_1__jordan2c,fc1_subset_1,fc4_pre_topc,t3_subset,d8_euclid,e6_96_1_1_1__jordan2c,t15_jordan2c]), [interesting(0.35),file(jordan2c,e7_96_1_1_1__jordan2c),[file(jordan2c,e7_96_1_1_1__jordan2c)]]). 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(0.9),axiom,file(connsp_1,t38_connsp_1)]). fof(e11_96_1_1_1__jordan2c,plain,( r1_tarski(c10_96_1_1__jordan2c,c2_96_1__jordan2c) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_c3_96_1_1__jordan2c,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k9_metric_1,dt_l1_metric_1,dt_u1_pre_topc,dt_c1_96_1_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1_1__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,d1_euclid,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,reflexivity_r1_tarski,symmetry_r1_xboole_0,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c10_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,dt_c3_96_1__jordan2c,de_c10_96_1_1__jordan2c,fc1_subset_1,fc4_pre_topc,t3_subset,d8_euclid,e10_96_1_1_1__jordan2c,e3_96_1__jordan2c,e7_96_1_1_1__jordan2c,t38_connsp_1]), [interesting(0.35),file(jordan2c,e11_96_1_1_1__jordan2c),[file(jordan2c,e11_96_1_1_1__jordan2c)]]). fof(e3_96_1_1_1__jordan2c,plain, ( r2_hidden(c1_96_1_1_1__jordan2c,k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c)) & ~ r2_hidden(c1_96_1_1_1__jordan2c,c4_96_1__jordan2c) ), inference(consider,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1_1__jordan2c])],[dh_c1_96_1_1_1__jordan2c,e2_96_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,e3_96_1_1_1__jordan2c),[file(jordan2c,e3_96_1_1_1__jordan2c)]]). fof(e12_96_1_1_1__jordan2c,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1_1__jordan2c])],[free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_finseq_2,existence_m1_pre_topc,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_finseq_2,dt_m1_pre_topc,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_struct_0,fc2_finseq_1,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc4_pre_topc,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_pre_topc,rc1_xreal_0,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_pre_topc,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_pre_topc,rc4_xreal_0,rc5_pre_topc,rc5_struct_0,free_g1_metric_1,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k15_euclid,dt_k1_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_k5_numbers,dt_l1_metric_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c3_96_1_1__jordan2c,dt_c9_96_1_1__jordan2c,de_c9_96_1_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_metric_1,rc1_subset_1,rc2_subset_1,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d1_euclid,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k14_euclid,dt_k9_metric_1,dt_c10_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c1_96_1_1_1__jordan2c,dt_c2_96_1__jordan2c,dt_c4_96_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c10_96_1_1__jordan2c,de_c4_96_1__jordan2c,de_c4_96_1_1__jordan2c,t1_subset,t3_subset,t7_boole,d7_euclid,e11_96_1_1_1__jordan2c,e3_96_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,e12_96_1_1_1__jordan2c),[file(jordan2c,e12_96_1_1_1__jordan2c)]]). fof(i2_96_1_1_1__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i2_96_1_1_1__jordan2c)]), [interesting(0.35),trivial,file(jordan2c,i2_96_1_1_1__jordan2c)]). fof(i1_96_1_1_1__jordan2c,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c,e1_96_1_1_1__jordan2c])],[e12_96_1_1_1__jordan2c,i2_96_1_1_1__jordan2c]), [interesting(0.35),file(jordan2c,i1_96_1_1_1__jordan2c),[file(jordan2c,i1_96_1_1_1__jordan2c)]]). fof(e17_96_1_1__jordan2c,plain,( r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,c4_96_1_1__jordan2c),c4_96_1__jordan2c) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c]),discharge_asm(discharge,[e1_96_1_1_1__jordan2c])],[e1_96_1_1_1__jordan2c,i1_96_1_1_1__jordan2c]), [interesting(0.5),file(jordan2c,e17_96_1_1__jordan2c),[file(jordan2c,e17_96_1_1__jordan2c)]]). fof(e18_96_1_1__jordan2c,plain,( ? [A] : ( v1_xreal_0(A) & ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),c4_96_1__jordan2c) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_pre_topc,dt_m1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_pre_topc,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_pre_topc,rc3_tbsp_1,rc4_funct_1,rc4_pre_topc,rc5_pre_topc,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k3_pre_topc,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pre_topc,fc4_pcomps_1,fc4_pre_topc,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc5_struct_0,t1_subset,t2_boole,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,free_g1_metric_1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_metric_1,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k1_zfmisc_1,dt_k3_xboole_0,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_metric_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_subset_1,dt_m2_subset_1,dt_c2_96_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_subset_1,fc2_euclid,fc2_membered,fc2_topreal1,fc3_pcomps_1,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_subset_1,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,d1_euclid,commutativity_k5_subset_1,idempotence_k5_subset_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k5_subset_1,dt_k9_metric_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c1_96_1_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1_1__jordan2c,dt_c3_96_1_1__jordan2c,dt_c4_96_1__jordan2c,dt_c4_96_1_1__jordan2c,de_c4_96_1__jordan2c,de_c4_96_1_1__jordan2c,cc2_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,t3_subset,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e17_96_1_1__jordan2c,e7_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,e18_96_1_1__jordan2c),[file(jordan2c,e18_96_1_1__jordan2c)]]). fof(i3_96_1_1__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i3_96_1_1__jordan2c)]), [interesting(0.5),trivial,file(jordan2c,i3_96_1_1__jordan2c)]). fof(i2_96_1_1__jordan2c,plain,( ? [A] : ( v1_xreal_0(A) & ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),c4_96_1__jordan2c) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,e1_96_1_1__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[e18_96_1_1__jordan2c,i3_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,i2_96_1_1__jordan2c),[file(jordan2c,i2_96_1_1__jordan2c)]]). fof(i1_96_1_1__jordan2c,plain,( ~ ( r2_hidden(c1_96_1_1__jordan2c,c4_96_1__jordan2c) & ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),c4_96_1__jordan2c) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_96_1_1__jordan2c,e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c]),discharge_asm(discharge,[e1_96_1_1__jordan2c])],[e1_96_1_1__jordan2c,i2_96_1_1__jordan2c]), [interesting(0.5),file(jordan2c,i1_96_1_1__jordan2c),[file(jordan2c,i1_96_1_1__jordan2c)]]). fof(i1_96_1_1_tmp__jordan2c,plain, ( m1_subset_1(c1_96_1_1__jordan2c,u1_struct_0(k14_euclid(c1_96__jordan2c))) => ~ ( r2_hidden(c1_96_1_1__jordan2c,c4_96_1__jordan2c) & ! [A] : ( v1_xreal_0(A) => ~ ( ~ r1_xreal_0(A,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),c1_96_1_1__jordan2c,A),c4_96_1__jordan2c) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c]),discharge_asm(discharge,[dt_c1_96_1_1__jordan2c])],[dt_c1_96_1_1__jordan2c,i1_96_1_1__jordan2c]), [interesting(0.65),e12_96_1__jordan2c]). fof(e12_96_1__jordan2c,plain,( ! [A] : ( m1_subset_1(A,u1_struct_0(k14_euclid(c1_96__jordan2c))) => ~ ( r2_hidden(A,c4_96_1__jordan2c) & ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(B,0) & r1_tarski(k9_metric_1(k14_euclid(c1_96__jordan2c),A,B),c4_96_1__jordan2c) ) ) ) ) ), inference(let,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[i1_96_1_1_tmp__jordan2c,dh_c1_96_1_1__jordan2c]), [interesting(0.65),file(jordan2c,e12_96_1__jordan2c),[file(jordan2c,e12_96_1__jordan2c)]]). fof(e13_96_1__jordan2c,plain,( v3_pre_topc(c4_96_1__jordan2c,k15_euclid(c1_96__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[existence_m1_pre_topc,dt_m1_pre_topc,cc1_arytm_3,cc1_finseq_1,cc1_pre_topc,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_pre_topc,rc3_tbsp_1,rc4_funct_1,rc4_pre_topc,rc5_pre_topc,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_pre_topc,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k3_pre_topc,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,dt_u1_pre_topc,dt_c2_96__jordan2c,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc2_finseq_1,fc3_pre_topc,fc4_pre_topc,fc4_subset_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc2_pcomps_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,free_g1_metric_1,abstractness_v1_metric_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_g1_metric_1,dt_k13_euclid,dt_k1_euclid,dt_k1_numbers,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,dt_c2_96_1__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_metric_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_metric_1,rc2_pre_topc,rc2_subset_1,rc3_metric_1,rc3_struct_0,rc5_struct_0,t1_numerals,t1_real,t2_subset,t4_real,t5_subset,t6_boole,t8_boole,d1_euclid,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_l1_metric_1,existence_m1_subset_1,dt_k14_euclid,dt_k15_euclid,dt_k1_zfmisc_1,dt_k5_pcomps_1,dt_k9_metric_1,dt_l1_metric_1,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c4_96_1__jordan2c,de_c4_96_1__jordan2c,cc2_xreal_0,fc1_subset_1,fc3_pcomps_1,fc4_pcomps_1,rqLessOrEqual__r1_xreal_0__r0_r0,t1_subset,t3_subset,t4_subset,t7_boole,d7_euclid,d8_euclid,spc0_numerals,spc0_boole,e12_96_1__jordan2c,t22_topmetr]), [interesting(0.65),file(jordan2c,e13_96_1__jordan2c),[file(jordan2c,e13_96_1__jordan2c)]]). fof(e17_96_1__jordan2c,plain,( v3_pre_topc(c2_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_finseq_1,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,cc2_funct_1,cc2_xreal_0,cc7_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_finseq_1,rc1_funct_1,rc1_jordan2c,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc2_tbsp_1,rc3_finseq_1,rc3_funct_1,rc3_tbsp_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,free_g1_metric_1,free_g1_pre_topc,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_finset_1,cc2_membered,cc2_tbsp_1,cc3_arytm_3,cc3_membered,cc4_membered,fc10_finset_1,fc11_finset_1,fc1_struct_0,fc27_membered,fc28_membered,fc29_membered,fc2_finseq_1,fc2_pre_topc,fc30_membered,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc3_pcomps_1,fc3_pre_topc,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_metric_1,rc2_metric_1,rc2_pcomps_1,rc2_pre_topc,rc3_finset_1,rc3_metric_1,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_struct_0,rc6_pre_topc,rc7_pre_topc,t1_subset,t2_boole,t4_subset,t5_subset,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_funct_1,cc1_pre_topc,cc6_membered,cc9_membered,fc1_euclid,fc1_jordan2c,fc2_euclid,fc2_membered,fc2_topreal1,fc4_pre_topc,fc5_pre_topc,rc1_pre_topc,rc1_subset_1,rc2_subset_1,rc3_pre_topc,rc5_pre_topc,t2_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,existence_l1_pre_topc,existence_m1_pre_topc,existence_m1_subset_1,redefinition_k5_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k2_pre_topc,dt_k3_pre_topc,dt_k3_xboole_0,dt_k5_subset_1,dt_l1_pre_topc,dt_m1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,dt_c4_96_1__jordan2c,de_c4_96_1__jordan2c,fc1_subset_1,t3_subset,d8_euclid,e16_96_1__jordan2c,e4_96_1__jordan2c,e13_96_1__jordan2c,t32_tops_2]), [interesting(0.65),file(jordan2c,e17_96_1__jordan2c),[file(jordan2c,e17_96_1__jordan2c)]]). fof(i4_96_1__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i4_96_1__jordan2c)]), [interesting(0.65),trivial,file(jordan2c,i4_96_1__jordan2c)]). fof(i3_96_1__jordan2c,plain,( v3_pre_topc(c2_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(conclusion,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c,e1_96_1__jordan2c])],[e17_96_1__jordan2c,i4_96_1__jordan2c]), [interesting(0.65),file(jordan2c,i3_96_1__jordan2c),[file(jordan2c,i3_96_1__jordan2c)]]). fof(i2_96_1__jordan2c,plain, ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,c2_96_1__jordan2c) ) => v3_pre_topc(c2_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(discharge_asm,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c,dt_c2_96_1__jordan2c]),discharge_asm(discharge,[e1_96_1__jordan2c])],[e1_96_1__jordan2c,i3_96_1__jordan2c]), [interesting(0.65),file(jordan2c,i2_96_1__jordan2c),[file(jordan2c,i2_96_1__jordan2c)]]). fof(i2_96_1_tmp__jordan2c,plain, ( m1_subset_1(c2_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,c2_96_1__jordan2c) ) => v3_pre_topc(c2_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ), inference(discharge_asm,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c]),discharge_asm(discharge,[dt_c2_96_1__jordan2c])],[dt_c2_96_1__jordan2c,i2_96_1__jordan2c]), [interesting(0.65),i1_96_1__jordan2c]). fof(i1_96_1__jordan2c,plain,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,A) ) => v3_pre_topc(A,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) ), inference(let,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c1_96_1__jordan2c,dt_c2_96__jordan2c])],[i2_96_1_tmp__jordan2c,dh_c2_96_1__jordan2c]), [interesting(0.65),file(jordan2c,i1_96_1__jordan2c),[file(jordan2c,i1_96_1__jordan2c)]]). fof(i1_96_1_tmp__jordan2c,plain, ( ( ~ v1_xboole_0(c1_96_1__jordan2c) & m1_subset_1(c1_96_1__jordan2c,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(c1_96_1__jordan2c,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),c1_96_1__jordan2c,A) ) => v3_pre_topc(A,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) ), inference(discharge_asm,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c2_96__jordan2c]),discharge_asm(discharge,[dt_c1_96_1__jordan2c])],[dt_c1_96_1__jordan2c,i1_96_1__jordan2c]), [interesting(0.8),e2_96__jordan2c]). fof(e2_96__jordan2c,plain,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)))) => ( ( v3_pre_topc(A,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) & r4_connsp_1(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c),A,B) ) => v3_pre_topc(B,k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ) ) ), inference(let,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c2_96__jordan2c])],[i1_96_1_tmp__jordan2c,dh_c1_96_1__jordan2c]), [interesting(0.8),file(jordan2c,e2_96__jordan2c),[file(jordan2c,e2_96__jordan2c)]]). 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(0.9),axiom,file(connsp_2,t24_connsp_2)]). fof(e3_96__jordan2c,plain,( v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(mizar_by,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c2_96__jordan2c])],[existence_m1_finseq_2,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_arytm_3,cc1_relset_1,cc2_arytm_3,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc14_finset_1,fc4_subset_1,rc1_arytm_3,rc1_jordan2c,rc2_finseq_1,rc2_finset_1,rc2_tbsp_1,rc2_xreal_0,rc3_tbsp_1,rc3_xreal_0,rc4_xreal_0,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k5_ordinal2,dt_l1_metric_1,dt_u1_pre_topc,cc1_finseq_1,cc1_xreal_0,cc2_funct_1,cc2_tbsp_1,cc2_xreal_0,cc3_arytm_3,cc7_xreal_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,rc1_finseq_1,rc1_funct_1,rc1_metric_1,rc1_xreal_0,rc2_funct_1,rc2_metric_1,rc2_pcomps_1,rc3_finseq_1,rc3_funct_1,rc3_metric_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,d1_euclid,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_struct_0,existence_m1_pre_topc,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_euclid,dt_k1_numbers,dt_k1_xboole_0,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_struct_0,dt_m1_pre_topc,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_pre_topc,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc1_euclid,fc1_struct_0,fc2_euclid,fc2_finseq_1,fc2_membered,fc2_topreal1,fc6_membered,rc1_finset_1,rc1_membered,rc1_pre_topc,rc2_pre_topc,rc3_finset_1,rc3_pre_topc,rc3_struct_0,rc4_finset_1,rc4_pre_topc,rc5_pre_topc,rc5_struct_0,t1_subset,t4_subset,t5_subset,t8_boole,d7_euclid,existence_l1_pre_topc,existence_m1_subset_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k3_pre_topc,dt_l1_pre_topc,dt_m1_subset_1,dt_u1_struct_0,dt_c1_96__jordan2c,dt_c2_96__jordan2c,cc15_membered,cc1_finset_1,cc1_funct_1,fc1_subset_1,fc3_pre_topc,fc4_pre_topc,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t6_boole,t7_boole,d8_euclid,e2_96__jordan2c,t24_connsp_2]), [interesting(0.8),file(jordan2c,e3_96__jordan2c),[file(jordan2c,e3_96__jordan2c)]]). fof(i4_96__jordan2c,theorem,( $true ), introduced(tautology,[file(jordan2c,i4_96__jordan2c)]), [interesting(0.8),trivial,file(jordan2c,i4_96__jordan2c)]). fof(i3_96__jordan2c,plain,( v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(conclusion,[status(thm),assumptions([e1_96__jordan2c,dt_c1_96__jordan2c,dt_c2_96__jordan2c])],[e3_96__jordan2c,i4_96__jordan2c]), [interesting(0.8),file(jordan2c,i3_96__jordan2c),[file(jordan2c,i3_96__jordan2c)]]). fof(i2_96__jordan2c,plain, ( v3_pre_topc(c2_96__jordan2c,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_96__jordan2c,dt_c2_96__jordan2c]),discharge_asm(discharge,[e1_96__jordan2c])],[e1_96__jordan2c,i3_96__jordan2c]), [interesting(0.8),file(jordan2c,i2_96__jordan2c),[file(jordan2c,i2_96__jordan2c)]]). fof(i2_96_tmp__jordan2c,plain, ( ( ~ v1_xboole_0(c2_96__jordan2c) & m1_subset_1(c2_96__jordan2c,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ) => ( v3_pre_topc(c2_96__jordan2c,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),c2_96__jordan2c)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_96__jordan2c]),discharge_asm(discharge,[dt_c2_96__jordan2c])],[dt_c2_96__jordan2c,i2_96__jordan2c]), [interesting(0.8),i1_96__jordan2c]). fof(i1_96__jordan2c,plain,( ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ) => ( v3_pre_topc(A,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),A)) ) ) ), inference(let,[status(thm),assumptions([dt_c1_96__jordan2c])],[i2_96_tmp__jordan2c,dh_c2_96__jordan2c]), [interesting(0.8),file(jordan2c,i1_96__jordan2c),[file(jordan2c,i1_96__jordan2c)]]). fof(i1_96_tmp__jordan2c,plain, ( m2_subset_1(c1_96__jordan2c,k1_numbers,k5_numbers) => ! [A] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(c1_96__jordan2c)))) ) => ( v3_pre_topc(A,k15_euclid(c1_96__jordan2c)) => v1_connsp_2(k3_pre_topc(k15_euclid(c1_96__jordan2c),A)) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_96__jordan2c])],[dt_c1_96__jordan2c,i1_96__jordan2c]), [interesting(1),t89_jordan2c]). 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(let,[status(thm),assumptions([])],[i1_96_tmp__jordan2c,dh_c1_96__jordan2c]), [interesting(1),file(jordan2c,t89_jordan2c),[file(jordan2c,t89_jordan2c)]]).