% Mizar ND problem: t19_topreal1,topreal1,828,66 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_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). 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_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(fc15_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc15_xreal_0)]). fof(fc16_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc16_xreal_0)]). fof(fc17_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc17_xreal_0)]). fof(fc18_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc18_xreal_0)]). fof(fc19_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc19_xreal_0)]). fof(fc20_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc20_xreal_0)]). fof(fc4_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), [interesting(0.9),axiom,file(int_1,fc4_int_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(fc5_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc5_xreal_0)]). fof(fc8_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), [interesting(0.9),axiom,file(int_1,fc8_int_1)]). fof(fc9_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), [interesting(0.9),axiom,file(int_1,fc9_int_1)]). fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,( k6_xcmplx_0(0,0) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]). fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,( k6_xcmplx_0(1,0) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealDiff__k6_xcmplx_0__r2_r0_r2,theorem,( k6_xcmplx_0(2,0) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2)]). fof(rqRealDiff__k6_xcmplx_0__r2_r1_r1,theorem,( k6_xcmplx_0(2,1) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r2_r2_r0,theorem,( k6_xcmplx_0(2,2) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). 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(commutativity_k17_euclid,theorem,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k17_euclid(A,B,C) = k17_euclid(A,C,B) ) ), file(euclid,k17_euclid), [interesting(0.9),axiom,file(euclid,k17_euclid)]). 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(redefinition_k5_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), [interesting(0.9),axiom,file(real_1,k5_real_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_k17_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k17_euclid(A,B,C),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k17_euclid), [interesting(0.9),axiom,file(euclid,k17_euclid)]). fof(dt_k18_euclid,axiom,( ! [A,B,C] : ( ( v1_xreal_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) ) => m1_subset_1(k18_euclid(A,B,C),u1_struct_0(k15_euclid(B))) ) ), file(euclid,k18_euclid), [interesting(0.9),axiom,file(euclid,k18_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_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), [interesting(0.9),axiom,file(real_1,k5_real_1)]). 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(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(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(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(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(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(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_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_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(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_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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_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_l1_struct_0,axiom,( $true ), file(struct_0,l1_struct_0), [interesting(0.9),axiom,file(struct_0,l1_struct_0)]). 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_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(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_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(cc4_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), [interesting(0.9),axiom,file(int_1,cc4_int_1)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). 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(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(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(fc1_xboole_0,theorem,( v1_xboole_0(k1_xboole_0) ), file(xboole_0,fc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]). 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(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_int_1,theorem,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), [interesting(0.9),axiom,file(int_1,rc1_int_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_nat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]). 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(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_int_1,theorem,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), [interesting(0.9),axiom,file(int_1,rc2_int_1)]). fof(rc2_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_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(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(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_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), [interesting(0.9),axiom,file(nat_1,rc3_nat_1)]). 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(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(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(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(fraenkel_a_3_0_topreal1,definition,( ! [A,B,C,D] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m1_subset_1(C,u1_struct_0(k15_euclid(B))) & m1_subset_1(D,u1_struct_0(k15_euclid(B))) ) => ( r2_hidden(A,a_3_0_topreal1(B,C,D)) <=> ? [E] : ( m1_subset_1(E,k1_numbers) & A = k17_euclid(B,k18_euclid(k5_real_1(1,E),B,C),k18_euclid(E,B,D)) & r1_xreal_0(0,E) & r1_xreal_0(E,1) ) ) ) ), file(topreal1,a_3_0_topreal1), [interesting(0.9),axiom,file(topreal1,a_3_0_topreal1)]). 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(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(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(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(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_k10_finseq_1,axiom,( $true ), file(finseq_1,k10_finseq_1), [interesting(0.9),axiom,file(finseq_1,k10_finseq_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_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k1_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k1_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k1_topreal1), [interesting(0.9),axiom,file(topreal1,k1_topreal1)]). 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_k21_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k21_euclid(A),k1_numbers) ) ), file(euclid,k21_euclid), [interesting(0.9),axiom,file(euclid,k21_euclid)]). fof(dt_k22_euclid,axiom,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => m1_subset_1(k22_euclid(A),k1_numbers) ) ), file(euclid,k22_euclid), [interesting(0.9),axiom,file(euclid,k22_euclid)]). 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_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). 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(dt_u1_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_struct_0)]). 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_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), [interesting(0.9),axiom,file(nat_1,cc1_nat_1)]). fof(cc2_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_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_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(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(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(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(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_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(fc1_topreal1,theorem,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => ~ v1_xboole_0(k1_topreal1(A,B,C)) ) ), file(topreal1,fc1_topreal1), [interesting(0.9),axiom,file(topreal1,fc1_topreal1)]). 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_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(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(rc1_xboole_0,theorem,( ? [A] : v1_xboole_0(A) ), file(xboole_0,rc1_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]). 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(rc2_xboole_0,theorem,( ? [A] : ~ v1_xboole_0(A) ), file(xboole_0,rc2_xboole_0), [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]). 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(rqLessOrEqual__r1_xreal_0__r0_r1,theorem,( r1_xreal_0(0,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1)]). fof(rqLessOrEqual__r1_xreal_0__r0_r2,theorem,( r1_xreal_0(0,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2)]). fof(rqLessOrEqual__r1_xreal_0__r1_r0,theorem,( ~ r1_xreal_0(1,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0)]). fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r2,theorem,( r1_xreal_0(1,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2)]). fof(rqLessOrEqual__r1_xreal_0__r2_r0,theorem,( ~ r1_xreal_0(2,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0)]). fof(rqLessOrEqual__r1_xreal_0__r2_r1,theorem,( ~ r1_xreal_0(2,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1)]). fof(rqLessOrEqual__r1_xreal_0__r2_r2,theorem,( r1_xreal_0(2,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2)]). 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(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(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(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(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(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(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(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(d3_topreal1,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ! [C] : ( m1_subset_1(C,u1_struct_0(k15_euclid(A))) => k1_topreal1(A,B,C) = a_3_0_topreal1(A,B,C) ) ) ) ), file(topreal1,d3_topreal1), [interesting(0.9),axiom,file(topreal1,d3_topreal1)]). fof(commutativity_k3_topreal1,theorem,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k3_topreal1(A,C,B) ) ), file(topreal1,k3_topreal1), [interesting(0.9),axiom,file(topreal1,k3_topreal1)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(redefinition_k3_topreal1,definition,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => k3_topreal1(A,B,C) = k1_topreal1(A,B,C) ) ), file(topreal1,k3_topreal1), [interesting(0.9),axiom,file(topreal1,k3_topreal1)]). fof(dt_k23_euclid,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => m1_subset_1(k23_euclid(A,B),u1_struct_0(k15_euclid(2))) ) ), file(euclid,k23_euclid), [interesting(0.9),axiom,file(euclid,k23_euclid)]). fof(dt_k3_topreal1,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) & m1_subset_1(C,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k3_topreal1(A,B,C),k1_zfmisc_1(u1_struct_0(k15_euclid(A)))) ) ), file(topreal1,k3_topreal1), [interesting(0.9),axiom,file(topreal1,k3_topreal1)]). 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(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_0_0_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_0_topreal1) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) & A = B & k21_euclid(B) = 0 & r1_xreal_0(k22_euclid(B),1) & r1_xreal_0(0,k22_euclid(B)) ) ) ), file(topreal1,a_0_0_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_0_topreal1)]). fof(d16_euclid,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => k23_euclid(A,B) = k10_finseq_1(A,B) ) ) ), file(euclid,d16_euclid), [interesting(0.9),axiom,file(euclid,d16_euclid)]). 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(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(dt_c1_24_1_2__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_1_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_1_2__topreal1)]). 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(dh_c1_24_1_2__topreal1,definition, ( ~ ( r2_hidden(c1_24_1_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) & ~ r2_hidden(c1_24_1_2__topreal1,a_0_0_topreal1) ) => ! [A] : ~ ( r2_hidden(A,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) & ~ r2_hidden(A,a_0_0_topreal1) ) ), introduced(definition,[new_symbol(c1_24_1_2__topreal1),file(topreal1,c1_24_1_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_1_2__topreal1)]). fof(e1_24_1_2__topreal1,assumption,( r2_hidden(c1_24_1_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ), introduced(assumption,[file(topreal1,e1_24_1_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_1_2__topreal1)]). fof(fc14_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc14_xreal_0)]). fof(fc3_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), [interesting(0.9),axiom,file(int_1,fc3_int_1)]). fof(fc13_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc13_xreal_0)]). fof(fc1_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc1_xreal_0)]). fof(fc5_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), [interesting(0.9),axiom,file(int_1,fc5_int_1)]). fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,( k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r0_r2_rm2,theorem,( k6_xcmplx_0(0,2) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm2_r2,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2)]). fof(rqRealDiff__k6_xcmplx_0__r1_r2_rm1,theorem,( k6_xcmplx_0(1,2) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r1_rm1_r2,theorem,( k6_xcmplx_0(1,k4_xcmplx_0(1)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1)]). fof(rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dh_c2_24_1_2__topreal1,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_1_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(0,0)),k18_euclid(A,2,k23_euclid(0,1))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) => ( m1_subset_1(c2_24_1_2__topreal1,k1_numbers) & c1_24_1_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))) & r1_xreal_0(0,c2_24_1_2__topreal1) & r1_xreal_0(c2_24_1_2__topreal1,1) ) ), introduced(definition,[new_symbol(c2_24_1_2__topreal1),file(topreal1,c2_24_1_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_1_2__topreal1)]). fof(fraenkel_a_0_10_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_10_topreal1) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & A = k17_euclid(2,k18_euclid(k5_real_1(1,B),2,k23_euclid(0,0)),k18_euclid(B,2,k23_euclid(0,1))) & r1_xreal_0(0,B) & r1_xreal_0(B,1) ) ) ), file(topreal1,a_0_10_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_10_topreal1)]). fof(e2_24_1_2__topreal1,plain,( r2_hidden(c1_24_1_2__topreal1,a_0_10_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_1_2__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_10_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_1_2__topreal1]), [interesting(0.5),file(topreal1,e2_24_1_2__topreal1),[file(topreal1,e2_24_1_2__topreal1)]]). fof(e3_24_1_2__topreal1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_1_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(0,0)),k18_euclid(A,2,k23_euclid(0,1))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_arithm,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_topreal1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k5_real_1,dt_m1_subset_1,dt_c1_24_1_2__topreal1,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_10_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_1_2__topreal1]), [interesting(0.5),file(topreal1,e3_24_1_2__topreal1),[file(topreal1,e3_24_1_2__topreal1)]]). fof(dt_c2_24_1_2__topreal1,plain,( m1_subset_1(c2_24_1_2__topreal1,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[dh_c2_24_1_2__topreal1,e3_24_1_2__topreal1]), [interesting(0.5),file(topreal1,c2_24_1_2__topreal1),[file(topreal1,c2_24_1_2__topreal1)]]). fof(rqLessOrEqual__r1_xreal_0__r0_rm1,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm2,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm1,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm2,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm1,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm2,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r2,theorem,( r1_xreal_0(k4_xcmplx_0(1),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm2,theorem,( ~ r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r0,theorem,( r1_xreal_0(k4_xcmplx_0(2),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r1,theorem,( r1_xreal_0(k4_xcmplx_0(2),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r2,theorem,( r1_xreal_0(k4_xcmplx_0(2),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm2,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(rqRealNeg__k4_xcmplx_0__r2_rm2,theorem,( k4_xcmplx_0(2) = k4_xcmplx_0(2) ), file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(rqRealNeg__k4_xcmplx_0__rm2_r2,theorem,( k4_xcmplx_0(k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2)]). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(dt_k2_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), [interesting(0.9),axiom,file(finseq_2,k2_finseq_2)]). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(redefinition_k4_finseqop,definition,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_finseqop(A,B,C) = k2_finseq_2(B,C) ) ), file(finseqop,k4_finseqop), [interesting(0.9),axiom,file(finseqop,k4_finseqop)]). fof(redefinition_m2_finseq_1,definition,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(dt_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_2(k4_finseqop(A,B,C),A,k4_finseq_2(B,A)) ) ), file(finseqop,k4_finseqop), [interesting(0.9),axiom,file(finseqop,k4_finseqop)]). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(redefinition_m2_finseq_2,definition,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(dt_k4_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k4_euclid(A),k1_numbers) ) ), file(euclid,k4_euclid), [interesting(0.9),axiom,file(euclid,k4_euclid)]). fof(dt_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(d4_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ), file(euclid,d4_euclid), [interesting(0.9),axiom,file(euclid,d4_euclid)]). fof(redefinition_k5_euclid,definition,( ! [A] : ( m1_subset_1(A,k5_numbers) => k5_euclid(A) = k4_euclid(A) ) ), file(euclid,k5_euclid), [interesting(0.9),axiom,file(euclid,k5_euclid)]). fof(dt_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ), file(euclid,k5_euclid), [interesting(0.9),axiom,file(euclid,k5_euclid)]). fof(dt_k16_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m1_subset_1(k16_euclid(A),u1_struct_0(k15_euclid(A))) ) ), file(euclid,k16_euclid), [interesting(0.9),axiom,file(euclid,k16_euclid)]). fof(d9_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k16_euclid(A) = k5_euclid(A) ) ), file(euclid,d9_euclid), [interesting(0.9),axiom,file(euclid,d9_euclid)]). fof(t32_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v1_xreal_0(B) => k18_euclid(B,A,k16_euclid(A)) = k16_euclid(A) ) ) ), file(euclid,t32_euclid), [interesting(0.9),axiom,file(euclid,t32_euclid)]). fof(t58_euclid,theorem,( k16_euclid(2) = k23_euclid(0,0) ), file(euclid,t58_euclid), [interesting(0.9),axiom,file(euclid,t58_euclid)]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(e1_24_1_2_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))) = k17_euclid(2,k16_euclid(2),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[existence_m1_finseq_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_finseq_2,dt_k2_zfmisc_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k4_finseq_2,dt_k4_finseqop,dt_l1_metric_1,dt_m1_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k14_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,d7_euclid,d1_euclid,d4_euclid,existence_m1_subset_1,redefinition_k5_euclid,dt_k10_finseq_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_subset_1,fc2_euclid,fc2_topreal1,fc5_int_1,fc5_membered,fc5_xreal_0,fc9_int_1,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_subset_1,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_m2_subset_1,dt_c2_24_1_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc2_membered,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t32_euclid,t58_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e1_24_1_2_1__topreal1),[file(topreal1,e1_24_1_2_1__topreal1)]]). fof(t31_euclid,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => ( k17_euclid(A,k16_euclid(A),B) = B & k17_euclid(A,B,k16_euclid(A)) = B ) ) ) ), file(euclid,t31_euclid), [interesting(0.9),axiom,file(euclid,t31_euclid)]). fof(e2_24_1_2_1__topreal1,plain,( k17_euclid(2,k16_euclid(2),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))) = k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k4_finseqop,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_l1_metric_1,dt_m2_finseq_2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_membered,rc1_metric_1,rc2_metric_1,rc3_metric_1,t1_subset,t4_subset,t5_subset,d1_euclid,d4_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k5_euclid,dt_k10_finseq_1,dt_k14_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc5_membered,rc1_int_1,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c2_24_1_2__topreal1,cc1_nat_1,cc2_int_1,cc2_nat_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_numerals,d8_euclid,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t31_euclid]), [interesting(0.35),file(topreal1,e2_24_1_2_1__topreal1),[file(topreal1,e2_24_1_2_1__topreal1)]]). fof(fc21_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc21_xreal_0)]). fof(fc22_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc22_xreal_0)]). fof(fc24_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc24_xreal_0)]). fof(fc2_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_1), [interesting(0.9),axiom,file(int_1,fc2_int_1)]). fof(fc2_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(nat_1,fc2_nat_1), [interesting(0.9),axiom,file(nat_1,fc2_nat_1)]). fof(fc7_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_1), [interesting(0.9),axiom,file(int_1,fc7_int_1)]). fof(fc23_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc23_xreal_0)]). fof(spc7_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(A,k3_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(t2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), [interesting(0.9),axiom,file(arithm,t2_arithm)]). fof(t3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), [interesting(0.9),axiom,file(arithm,t3_arithm)]). fof(commutativity_k3_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(commutativity_k4_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(redefinition_k4_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(dt_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(fc4_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc4_xreal_0)]). fof(rqRealMult__k3_xcmplx_0__r0_r0_r0,theorem,( k3_xcmplx_0(0,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r1_r0,theorem,( k3_xcmplx_0(0,1) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r2_r0,theorem,( k3_xcmplx_0(0,2) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r2_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r0_r0,theorem,( k3_xcmplx_0(1,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r2_r2,theorem,( k3_xcmplx_0(1,2) = 2 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r2_r2)]). fof(rqRealMult__k3_xcmplx_0__r2_r0_r0,theorem,( k3_xcmplx_0(2,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r2_r1_r2,theorem,( k3_xcmplx_0(2,1) = 2 ), file(arithm,rqRealMult__k3_xcmplx_0__r2_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r2_r1_r2)]). fof(t61_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) => k18_euclid(A,2,B) = k23_euclid(k3_xcmplx_0(A,k21_euclid(B)),k3_xcmplx_0(A,k22_euclid(B))) ) ) ), file(euclid,t61_euclid), [interesting(0.9),axiom,file(euclid,t61_euclid)]). fof(rqRealMult__k3_xcmplx_0__r1_r1_r1,theorem,( k3_xcmplx_0(1,1) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1)]). fof(e3_24_1_2_1__topreal1,plain,( k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1)) = k23_euclid(k4_real_1(c2_24_1_2__topreal1,k21_euclid(k23_euclid(0,1))),k4_real_1(c2_24_1_2__topreal1,k22_euclid(k23_euclid(0,1)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m1_subset_1,redefinition_k4_real_1,dt_k15_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_1_2__topreal1,cc2_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e3_24_1_2_1__topreal1),[file(topreal1,e3_24_1_2_1__topreal1)]]). fof(t56_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( k21_euclid(k23_euclid(A,B)) = A & k22_euclid(k23_euclid(A,B)) = B ) ) ) ), file(euclid,t56_euclid), [interesting(0.9),axiom,file(euclid,t56_euclid)]). fof(e1_24__topreal1,plain, ( k21_euclid(k23_euclid(0,1)) = 0 & k22_euclid(k23_euclid(0,1)) = 1 ), inference(mizar_by,[status(thm),assumptions([])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,cc2_xreal_0,d16_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t56_euclid]), [interesting(0.8),file(topreal1,e1_24__topreal1),[file(topreal1,e1_24__topreal1)]]). fof(e4_24_1_2_1__topreal1,plain,( k23_euclid(k4_real_1(c2_24_1_2__topreal1,k21_euclid(k23_euclid(0,1))),k4_real_1(c2_24_1_2__topreal1,k22_euclid(k23_euclid(0,1)))) = k23_euclid(0,c2_24_1_2__topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc4_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc2_boole,spc2_numerals,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,commutativity_k3_xcmplx_0,commutativity_k4_real_1,redefinition_k4_real_1,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_c2_24_1_2__topreal1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,d16_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_24__topreal1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e4_24_1_2_1__topreal1),[file(topreal1,e4_24_1_2_1__topreal1)]]). fof(e5_24_1_2__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))) = k23_euclid(0,c2_24_1_2__topreal1) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[e1_24_1_2_1__topreal1,e2_24_1_2_1__topreal1,e3_24_1_2_1__topreal1,e4_24_1_2_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_1_2__topreal1),[file(topreal1,e5_24_1_2__topreal1)]]). fof(e4_24_1_2__topreal1,plain, ( c1_24_1_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))) & r1_xreal_0(0,c2_24_1_2__topreal1) & r1_xreal_0(c2_24_1_2__topreal1,1) ), inference(consider,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[dh_c2_24_1_2__topreal1,e3_24_1_2__topreal1]), [interesting(0.5),file(topreal1,e4_24_1_2__topreal1),[file(topreal1,e4_24_1_2__topreal1)]]). fof(e6_24_1_2__topreal1,plain, ( k21_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1)))) = 0 & r1_xreal_0(k22_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1)))),1) & r1_xreal_0(0,k22_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_1_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_1_2__topreal1,2,k23_euclid(0,1))))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_1_2__topreal1,dt_c2_24_1_2__topreal1,cc2_xreal_0,fc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_1_2__topreal1,e4_24_1_2__topreal1,t56_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.5),file(topreal1,e6_24_1_2__topreal1),[file(topreal1,e6_24_1_2__topreal1)]]). fof(e7_24_1_2__topreal1,plain,( r2_hidden(c1_24_1_2__topreal1,a_0_0_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_1_2__topreal1,dt_c2_24_1_2__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_1_2__topreal1,e4_24_1_2__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2]), [interesting(0.5),file(topreal1,e7_24_1_2__topreal1),[file(topreal1,e7_24_1_2__topreal1)]]). fof(i3_24_1_2__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_1_2__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_1_2__topreal1)]). fof(i2_24_1_2__topreal1,plain,( r2_hidden(c1_24_1_2__topreal1,a_0_0_topreal1) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_1_2__topreal1,e1_24_1_2__topreal1])],[e7_24_1_2__topreal1,i3_24_1_2__topreal1]), [interesting(0.5),file(topreal1,i2_24_1_2__topreal1),[file(topreal1,i2_24_1_2__topreal1)]]). fof(i1_24_1_2__topreal1,plain,( ~ ( r2_hidden(c1_24_1_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) & ~ r2_hidden(c1_24_1_2__topreal1,a_0_0_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_1_2__topreal1]),discharge_asm(discharge,[e1_24_1_2__topreal1])],[e1_24_1_2__topreal1,i2_24_1_2__topreal1]), [interesting(0.5),file(topreal1,i1_24_1_2__topreal1),[file(topreal1,i1_24_1_2__topreal1)]]). fof(i1_24_1_2_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_1_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) & ~ r2_hidden(c1_24_1_2__topreal1,a_0_0_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_1_2__topreal1])],[dt_c1_24_1_2__topreal1,i1_24_1_2__topreal1]), [interesting(0.65),e2_24_1__topreal1]). fof(e2_24_1__topreal1,plain,( r1_tarski(k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1)),a_0_0_topreal1) ), inference(let,[status(thm),assumptions([])],[i1_24_1_2_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_0_topreal1,d3_tarski,dh_c1_24_1_2__topreal1]), [interesting(0.65),file(topreal1,e2_24_1__topreal1),[file(topreal1,e2_24_1__topreal1)]]). fof(dt_c1_24_1_1__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_1_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_1_1__topreal1)]). fof(dh_c1_24_1_1__topreal1,definition, ( ~ ( r2_hidden(c1_24_1_1__topreal1,a_0_0_topreal1) & ~ r2_hidden(c1_24_1_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ) => ! [A] : ~ ( r2_hidden(A,a_0_0_topreal1) & ~ r2_hidden(A,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ) ), introduced(definition,[new_symbol(c1_24_1_1__topreal1),file(topreal1,c1_24_1_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_1_1__topreal1)]). fof(e1_24_1_1__topreal1,assumption,( r2_hidden(c1_24_1_1__topreal1,a_0_0_topreal1) ), introduced(assumption,[file(topreal1,e1_24_1_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_1_1__topreal1)]). fof(dh_c2_24_1_1__topreal1,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_1_1__topreal1 = A & k21_euclid(A) = 0 & r1_xreal_0(k22_euclid(A),1) & r1_xreal_0(0,k22_euclid(A)) ) => ( m1_subset_1(c2_24_1_1__topreal1,u1_struct_0(k15_euclid(2))) & c1_24_1_1__topreal1 = c2_24_1_1__topreal1 & k21_euclid(c2_24_1_1__topreal1) = 0 & r1_xreal_0(k22_euclid(c2_24_1_1__topreal1),1) & r1_xreal_0(0,k22_euclid(c2_24_1_1__topreal1)) ) ), introduced(definition,[new_symbol(c2_24_1_1__topreal1),file(topreal1,c2_24_1_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_1_1__topreal1)]). fof(e2_24_1_1__topreal1,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_1_1__topreal1 = A & k21_euclid(A) = 0 & r1_xreal_0(k22_euclid(A),1) & r1_xreal_0(0,k22_euclid(A)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_1,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_k1_xboole_0,dt_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_m1_subset_1,dt_u1_struct_0,dt_c1_24_1_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_topreal1,d8_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_1_1__topreal1]), [interesting(0.5),file(topreal1,e2_24_1_1__topreal1),[file(topreal1,e2_24_1_1__topreal1)]]). fof(dt_c2_24_1_1__topreal1,plain,( m1_subset_1(c2_24_1_1__topreal1,u1_struct_0(k15_euclid(2))) ), inference(consider,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[dh_c2_24_1_1__topreal1,e2_24_1_1__topreal1]), [interesting(0.5),file(topreal1,c2_24_1_1__topreal1),[file(topreal1,c2_24_1_1__topreal1)]]). fof(e1_24_1_1_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k22_euclid(c2_24_1_1__topreal1)),2,k23_euclid(0,0)),k18_euclid(k22_euclid(c2_24_1_1__topreal1),2,k23_euclid(0,1))) = k17_euclid(2,k16_euclid(2),k18_euclid(k22_euclid(c2_24_1_1__topreal1),2,k23_euclid(0,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[existence_m1_finseq_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_finseq_2,dt_k2_zfmisc_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k4_finseq_2,dt_k4_finseqop,dt_l1_metric_1,dt_m1_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k14_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,d7_euclid,d1_euclid,d4_euclid,existence_m1_subset_1,redefinition_k5_euclid,dt_k10_finseq_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_subset_1,fc2_euclid,fc2_topreal1,fc5_int_1,fc5_membered,fc5_xreal_0,fc9_int_1,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_subset_1,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_m2_subset_1,dt_c2_24_1_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc2_membered,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t32_euclid,t58_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e1_24_1_1_1__topreal1),[file(topreal1,e1_24_1_1_1__topreal1)]]). fof(e2_24_1_1_1__topreal1,plain,( k17_euclid(2,k16_euclid(2),k18_euclid(k22_euclid(c2_24_1_1__topreal1),2,k23_euclid(0,1))) = k18_euclid(k22_euclid(c2_24_1_1__topreal1),2,k23_euclid(0,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k4_finseqop,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_l1_metric_1,dt_m2_finseq_2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_membered,rc1_metric_1,rc2_metric_1,rc3_metric_1,t1_subset,t4_subset,t5_subset,d1_euclid,d4_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k5_euclid,dt_k10_finseq_1,dt_k14_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc5_membered,rc1_int_1,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k22_euclid,dt_k23_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c2_24_1_1__topreal1,cc1_nat_1,cc2_int_1,cc2_nat_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_numerals,d8_euclid,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t31_euclid]), [interesting(0.35),file(topreal1,e2_24_1_1_1__topreal1),[file(topreal1,e2_24_1_1_1__topreal1)]]). fof(e3_24_1_1_1__topreal1,plain,( k18_euclid(k22_euclid(c2_24_1_1__topreal1),2,k23_euclid(0,1)) = k23_euclid(k4_real_1(k22_euclid(c2_24_1_1__topreal1),k21_euclid(k23_euclid(0,1))),k4_real_1(k22_euclid(c2_24_1_1__topreal1),k22_euclid(k23_euclid(0,1)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m1_subset_1,redefinition_k4_real_1,dt_k15_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_1_1__topreal1,cc2_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e3_24_1_1_1__topreal1),[file(topreal1,e3_24_1_1_1__topreal1)]]). fof(e4_24_1_1__topreal1,plain, ( k21_euclid(c2_24_1_1__topreal1) = 0 & r1_xreal_0(k22_euclid(c2_24_1_1__topreal1),1) & r1_xreal_0(0,k22_euclid(c2_24_1_1__topreal1)) ), inference(consider,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[dh_c2_24_1_1__topreal1,e2_24_1_1__topreal1]), [interesting(0.5),file(topreal1,e4_24_1_1__topreal1),[file(topreal1,e4_24_1_1__topreal1)]]). fof(t57_euclid,theorem,( ! [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) => A = k23_euclid(k21_euclid(A),k22_euclid(A)) ) ), file(euclid,t57_euclid), [interesting(0.9),axiom,file(euclid,t57_euclid)]). fof(e4_24_1_1_1__topreal1,plain,( k23_euclid(k4_real_1(k22_euclid(c2_24_1_1__topreal1),k21_euclid(k23_euclid(0,1))),k4_real_1(k22_euclid(c2_24_1_1__topreal1),k22_euclid(k23_euclid(0,1)))) = c2_24_1_1__topreal1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc4_xreal_0,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc7_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k4_real_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_1_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24__topreal1,e4_24_1_1__topreal1,t57_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e4_24_1_1_1__topreal1),[file(topreal1,e4_24_1_1_1__topreal1)]]). fof(e5_24_1_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k22_euclid(c2_24_1_1__topreal1)),2,k23_euclid(0,0)),k18_euclid(k22_euclid(c2_24_1_1__topreal1),2,k23_euclid(0,1))) = c2_24_1_1__topreal1 ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[e1_24_1_1_1__topreal1,e2_24_1_1_1__topreal1,e3_24_1_1_1__topreal1,e4_24_1_1_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_1_1__topreal1),[file(topreal1,e5_24_1_1__topreal1)]]). fof(e3_24_1_1__topreal1,plain,( c1_24_1_1__topreal1 = c2_24_1_1__topreal1 ), inference(consider,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[dh_c2_24_1_1__topreal1,e2_24_1_1__topreal1]), [interesting(0.5),file(topreal1,e3_24_1_1__topreal1),[file(topreal1,e3_24_1_1__topreal1)]]). fof(e6_24_1_1__topreal1,plain,( r2_hidden(c1_24_1_1__topreal1,a_0_10_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_1_1__topreal1,dt_c2_24_1_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_10_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_1_1__topreal1,e3_24_1_1__topreal1,e4_24_1_1__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.5),file(topreal1,e6_24_1_1__topreal1),[file(topreal1,e6_24_1_1__topreal1)]]). fof(e7_24_1_1__topreal1,plain,( r2_hidden(c1_24_1_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_1_1__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_10_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_1_1__topreal1]), [interesting(0.5),file(topreal1,e7_24_1_1__topreal1),[file(topreal1,e7_24_1_1__topreal1)]]). fof(i3_24_1_1__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_1_1__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_1_1__topreal1)]). fof(i2_24_1_1__topreal1,plain,( r2_hidden(c1_24_1_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_1_1__topreal1,e1_24_1_1__topreal1])],[e7_24_1_1__topreal1,i3_24_1_1__topreal1]), [interesting(0.5),file(topreal1,i2_24_1_1__topreal1),[file(topreal1,i2_24_1_1__topreal1)]]). fof(i1_24_1_1__topreal1,plain,( ~ ( r2_hidden(c1_24_1_1__topreal1,a_0_0_topreal1) & ~ r2_hidden(c1_24_1_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_1_1__topreal1]),discharge_asm(discharge,[e1_24_1_1__topreal1])],[e1_24_1_1__topreal1,i2_24_1_1__topreal1]), [interesting(0.5),file(topreal1,i1_24_1_1__topreal1),[file(topreal1,i1_24_1_1__topreal1)]]). fof(i1_24_1_1_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_1_1__topreal1,a_0_0_topreal1) & ~ r2_hidden(c1_24_1_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_1_1__topreal1])],[dt_c1_24_1_1__topreal1,i1_24_1_1__topreal1]), [interesting(0.65),e1_24_1__topreal1]). fof(e1_24_1__topreal1,plain,( r1_tarski(a_0_0_topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1))) ), inference(let,[status(thm),assumptions([])],[i1_24_1_1_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_0_topreal1,d3_tarski,dh_c1_24_1_1__topreal1]), [interesting(0.65),file(topreal1,e1_24_1__topreal1),[file(topreal1,e1_24_1__topreal1)]]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]). fof(e3_24_1__topreal1,plain,( a_0_0_topreal1 = k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1)) ), inference(mizar_by,[status(thm),assumptions([])],[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_k6_xcmplx_0,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc20_xreal_0,fc4_int_1,fc4_subset_1,fc5_xreal_0,fc8_int_1,fc9_int_1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t4_arithm,free_g1_metric_1,free_g1_pre_topc,commutativity_k17_euclid,abstractness_v1_metric_1,existence_l1_metric_1,redefinition_k5_real_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_euclid,dt_k5_real_1,dt_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t1_subset,t2_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,reflexivity_r1_tarski,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,t3_subset,t2_tarski,fraenkel_a_0_0_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_1__topreal1,e1_24_1__topreal1,d10_xboole_0]), [interesting(0.65),file(topreal1,e3_24_1__topreal1),[file(topreal1,e3_24_1__topreal1)]]). fof(i1_24_1__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i1_24_1__topreal1)]), [interesting(0.65),trivial,file(topreal1,i1_24_1__topreal1)]). fof(e4_24__topreal1,plain,( a_0_0_topreal1 = k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1)) ), inference(conclusion,[status(thm),assumptions([])],[e3_24_1__topreal1,i1_24_1__topreal1]), [interesting(0.8),file(topreal1,e4_24__topreal1),[file(topreal1,e4_24__topreal1)]]). fof(fraenkel_a_0_1_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_1_topreal1) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) & A = B & r1_xreal_0(k21_euclid(B),1) & r1_xreal_0(0,k21_euclid(B)) & k22_euclid(B) = 1 ) ) ), file(topreal1,a_0_1_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_1_topreal1)]). fof(dt_c1_24_2_2__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_2_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_2_2__topreal1)]). fof(dh_c1_24_2_2__topreal1,definition, ( ~ ( r2_hidden(c1_24_2_2__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) & ~ r2_hidden(c1_24_2_2__topreal1,a_0_1_topreal1) ) => ! [A] : ~ ( r2_hidden(A,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) & ~ r2_hidden(A,a_0_1_topreal1) ) ), introduced(definition,[new_symbol(c1_24_2_2__topreal1),file(topreal1,c1_24_2_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_2_2__topreal1)]). fof(e1_24_2_2__topreal1,assumption,( r2_hidden(c1_24_2_2__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ), introduced(assumption,[file(topreal1,e1_24_2_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_2_2__topreal1)]). fof(dh_c2_24_2_2__topreal1,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_2_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(0,1)),k18_euclid(A,2,k23_euclid(1,1))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) => ( m1_subset_1(c2_24_2_2__topreal1,k1_numbers) & c1_24_2_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))) & r1_xreal_0(0,c2_24_2_2__topreal1) & r1_xreal_0(c2_24_2_2__topreal1,1) ) ), introduced(definition,[new_symbol(c2_24_2_2__topreal1),file(topreal1,c2_24_2_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_2_2__topreal1)]). fof(fraenkel_a_0_11_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_11_topreal1) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & A = k17_euclid(2,k18_euclid(k5_real_1(1,B),2,k23_euclid(0,1)),k18_euclid(B,2,k23_euclid(1,1))) & r1_xreal_0(0,B) & r1_xreal_0(B,1) ) ) ), file(topreal1,a_0_11_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_11_topreal1)]). fof(e2_24_2_2__topreal1,plain,( r2_hidden(c1_24_2_2__topreal1,a_0_11_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_2_2__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_11_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_2_2__topreal1]), [interesting(0.5),file(topreal1,e2_24_2_2__topreal1),[file(topreal1,e2_24_2_2__topreal1)]]). fof(e3_24_2_2__topreal1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_2_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(0,1)),k18_euclid(A,2,k23_euclid(1,1))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_arithm,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_topreal1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k5_real_1,dt_m1_subset_1,dt_c1_24_2_2__topreal1,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_11_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_2_2__topreal1]), [interesting(0.5),file(topreal1,e3_24_2_2__topreal1),[file(topreal1,e3_24_2_2__topreal1)]]). fof(dt_c2_24_2_2__topreal1,plain,( m1_subset_1(c2_24_2_2__topreal1,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[dh_c2_24_2_2__topreal1,e3_24_2_2__topreal1]), [interesting(0.5),file(topreal1,c2_24_2_2__topreal1),[file(topreal1,c2_24_2_2__topreal1)]]). fof(spc2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,k4_xcmplx_0(1)) = k4_xcmplx_0(A) ) ), file(arithm,spc2_arithm), [interesting(0.9),axiom,file(arithm,spc2_arithm)]). fof(rqRealMult__k3_xcmplx_0__r0_rm2_r0,theorem,( k3_xcmplx_0(0,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_rm2_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_rm2_rm2,theorem,( k3_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealMult__k3_xcmplx_0__r1_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_rm2_rm2)]). fof(rqRealMult__k3_xcmplx_0__rm2_r0_r0,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__rm2_r1_rm2,theorem,( k3_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(2) ), file(arithm,rqRealMult__k3_xcmplx_0__rm2_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__rm2_r1_rm2)]). fof(e1_24_2_2_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,c2_24_2_2__topreal1),0),k4_real_1(k5_real_1(1,c2_24_2_2__topreal1),k22_euclid(k23_euclid(0,1)))),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_2_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24__topreal1,t61_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e1_24_2_2_1__topreal1),[file(topreal1,e1_24_2_2_1__topreal1)]]). fof(e2_24__topreal1,plain, ( k21_euclid(k23_euclid(1,1)) = 1 & k22_euclid(k23_euclid(1,1)) = 1 ), inference(mizar_by,[status(thm),assumptions([])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc2_boole,spc2_numerals,t2_subset,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,cc2_xreal_0,d16_euclid,spc1_numerals,spc1_boole,t56_euclid]), [interesting(0.8),file(topreal1,e2_24__topreal1),[file(topreal1,e2_24__topreal1)]]). fof(e2_24_2_2_1__topreal1,plain,( k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,c2_24_2_2__topreal1),0),k4_real_1(k5_real_1(1,c2_24_2_2__topreal1),k22_euclid(k23_euclid(0,1)))),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(0,k5_real_1(1,c2_24_2_2__topreal1)),k23_euclid(c2_24_2_2__topreal1,k4_real_1(c2_24_2_2__topreal1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_2_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24__topreal1,e2_24__topreal1,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e2_24_2_2_1__topreal1),[file(topreal1,e2_24_2_2_1__topreal1)]]). fof(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc1_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), [interesting(0.9),axiom,file(int_1,fc1_int_1)]). fof(fc1_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), [interesting(0.9),axiom,file(nat_1,fc1_nat_1)]). fof(fc3_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), [interesting(0.9),axiom,file(nat_1,fc3_nat_1)]). fof(fc4_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), [interesting(0.9),axiom,file(nat_1,fc4_nat_1)]). fof(fc6_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), [interesting(0.9),axiom,file(int_1,fc6_int_1)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). fof(spc1_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,k4_xcmplx_0(B)) = k6_xcmplx_0(A,B) ) ), file(arithm,spc1_arithm), [interesting(0.9),axiom,file(arithm,spc1_arithm)]). fof(spc5_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ), file(arithm,spc5_arithm), [interesting(0.9),axiom,file(arithm,spc5_arithm)]). fof(spc6_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k2_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(A,k2_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(t1_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), [interesting(0.9),axiom,file(arithm,t1_arithm)]). fof(commutativity_k2_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(commutativity_k3_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(redefinition_k3_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r0_r0,theorem,( k2_xcmplx_0(0,0) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r2_r2,theorem,( k2_xcmplx_0(0,2) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__r1_r1_r2,theorem,( k2_xcmplx_0(1,1) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r2_r0_r2,theorem,( k2_xcmplx_0(2,0) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm1_r1,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm2_r0,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r1_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),1) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r2_r1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),2) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r2_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),2) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0)]). fof(t60_euclid,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => k17_euclid(2,k23_euclid(A,B),k23_euclid(C,D)) = k23_euclid(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ), file(euclid,t60_euclid), [interesting(0.9),axiom,file(euclid,t60_euclid)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm1_r0,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0)]). fof(e3_24_2_2_1__topreal1,plain,( k17_euclid(2,k23_euclid(0,k5_real_1(1,c2_24_2_2__topreal1)),k23_euclid(c2_24_2_2__topreal1,k4_real_1(c2_24_2_2__topreal1,1))) = k23_euclid(k3_real_1(0,c2_24_2_2__topreal1),k3_real_1(k5_real_1(1,c2_24_2_2__topreal1),c2_24_2_2__topreal1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k17_euclid,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_c2_24_2_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t60_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0]), [interesting(0.35),file(topreal1,e3_24_2_2_1__topreal1),[file(topreal1,e3_24_2_2_1__topreal1)]]). fof(e4_24_2_2_1__topreal1,plain,( k23_euclid(k3_real_1(0,c2_24_2_2__topreal1),k3_real_1(k5_real_1(1,c2_24_2_2__topreal1),c2_24_2_2__topreal1)) = k23_euclid(c2_24_2_2__topreal1,1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_xreal_0,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc1_arithm,spc2_boole,spc2_numerals,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k5_real_1,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_c2_24_2_2__topreal1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,d16_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__r1_r0_r1]), [interesting(0.35),file(topreal1,e4_24_2_2_1__topreal1),[file(topreal1,e4_24_2_2_1__topreal1)]]). fof(e5_24_2_2__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))) = k23_euclid(c2_24_2_2__topreal1,1) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[e1_24_2_2_1__topreal1,e2_24_2_2_1__topreal1,e3_24_2_2_1__topreal1,e4_24_2_2_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_2_2__topreal1),[file(topreal1,e5_24_2_2__topreal1)]]). fof(e4_24_2_2__topreal1,plain, ( c1_24_2_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))) & r1_xreal_0(0,c2_24_2_2__topreal1) & r1_xreal_0(c2_24_2_2__topreal1,1) ), inference(consider,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[dh_c2_24_2_2__topreal1,e3_24_2_2__topreal1]), [interesting(0.5),file(topreal1,e4_24_2_2__topreal1),[file(topreal1,e4_24_2_2__topreal1)]]). fof(e6_24_2_2__topreal1,plain, ( r1_xreal_0(k21_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1)))),1) & r1_xreal_0(0,k21_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1))))) & k22_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_2_2__topreal1),2,k23_euclid(0,1)),k18_euclid(c2_24_2_2__topreal1,2,k23_euclid(1,1)))) = 1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_2_2__topreal1,dt_c2_24_2_2__topreal1,cc2_xreal_0,fc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_2_2__topreal1,e4_24_2_2__topreal1,t56_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.5),file(topreal1,e6_24_2_2__topreal1),[file(topreal1,e6_24_2_2__topreal1)]]). fof(e7_24_2_2__topreal1,plain,( r2_hidden(c1_24_2_2__topreal1,a_0_1_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_2_2__topreal1,dt_c2_24_2_2__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_1_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_2_2__topreal1,e4_24_2_2__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2]), [interesting(0.5),file(topreal1,e7_24_2_2__topreal1),[file(topreal1,e7_24_2_2__topreal1)]]). fof(i3_24_2_2__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_2_2__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_2_2__topreal1)]). fof(i2_24_2_2__topreal1,plain,( r2_hidden(c1_24_2_2__topreal1,a_0_1_topreal1) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_2_2__topreal1,e1_24_2_2__topreal1])],[e7_24_2_2__topreal1,i3_24_2_2__topreal1]), [interesting(0.5),file(topreal1,i2_24_2_2__topreal1),[file(topreal1,i2_24_2_2__topreal1)]]). fof(i1_24_2_2__topreal1,plain,( ~ ( r2_hidden(c1_24_2_2__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) & ~ r2_hidden(c1_24_2_2__topreal1,a_0_1_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_2_2__topreal1]),discharge_asm(discharge,[e1_24_2_2__topreal1])],[e1_24_2_2__topreal1,i2_24_2_2__topreal1]), [interesting(0.5),file(topreal1,i1_24_2_2__topreal1),[file(topreal1,i1_24_2_2__topreal1)]]). fof(i1_24_2_2_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_2_2__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) & ~ r2_hidden(c1_24_2_2__topreal1,a_0_1_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_2_2__topreal1])],[dt_c1_24_2_2__topreal1,i1_24_2_2__topreal1]), [interesting(0.65),e2_24_2__topreal1]). fof(e2_24_2__topreal1,plain,( r1_tarski(k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1)),a_0_1_topreal1) ), inference(let,[status(thm),assumptions([])],[i1_24_2_2_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_1_topreal1,d3_tarski,dh_c1_24_2_2__topreal1]), [interesting(0.65),file(topreal1,e2_24_2__topreal1),[file(topreal1,e2_24_2__topreal1)]]). fof(dt_c1_24_2_1__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_2_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_2_1__topreal1)]). fof(dh_c1_24_2_1__topreal1,definition, ( ~ ( r2_hidden(c1_24_2_1__topreal1,a_0_1_topreal1) & ~ r2_hidden(c1_24_2_1__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ) => ! [A] : ~ ( r2_hidden(A,a_0_1_topreal1) & ~ r2_hidden(A,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ) ), introduced(definition,[new_symbol(c1_24_2_1__topreal1),file(topreal1,c1_24_2_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_2_1__topreal1)]). fof(e1_24_2_1__topreal1,assumption,( r2_hidden(c1_24_2_1__topreal1,a_0_1_topreal1) ), introduced(assumption,[file(topreal1,e1_24_2_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_2_1__topreal1)]). fof(dh_c2_24_2_1__topreal1,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_2_1__topreal1 = A & r1_xreal_0(k21_euclid(A),1) & r1_xreal_0(0,k21_euclid(A)) & k22_euclid(A) = 1 ) => ( m1_subset_1(c2_24_2_1__topreal1,u1_struct_0(k15_euclid(2))) & c1_24_2_1__topreal1 = c2_24_2_1__topreal1 & r1_xreal_0(k21_euclid(c2_24_2_1__topreal1),1) & r1_xreal_0(0,k21_euclid(c2_24_2_1__topreal1)) & k22_euclid(c2_24_2_1__topreal1) = 1 ) ), introduced(definition,[new_symbol(c2_24_2_1__topreal1),file(topreal1,c2_24_2_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_2_1__topreal1)]). fof(e2_24_2_1__topreal1,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_2_1__topreal1 = A & r1_xreal_0(k21_euclid(A),1) & r1_xreal_0(0,k21_euclid(A)) & k22_euclid(A) = 1 ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_1,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_k1_xboole_0,dt_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_m1_subset_1,dt_u1_struct_0,dt_c1_24_2_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_1_topreal1,d8_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_2_1__topreal1]), [interesting(0.5),file(topreal1,e2_24_2_1__topreal1),[file(topreal1,e2_24_2_1__topreal1)]]). fof(dt_c2_24_2_1__topreal1,plain,( m1_subset_1(c2_24_2_1__topreal1,u1_struct_0(k15_euclid(2))) ), inference(consider,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[dh_c2_24_2_1__topreal1,e2_24_2_1__topreal1]), [interesting(0.5),file(topreal1,c2_24_2_1__topreal1),[file(topreal1,c2_24_2_1__topreal1)]]). fof(e1_24_2_1_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),2,k23_euclid(0,1)),k18_euclid(k21_euclid(c2_24_2_1__topreal1),2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),0),k4_real_1(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),k22_euclid(k23_euclid(0,1)))),k18_euclid(k21_euclid(c2_24_2_1__topreal1),2,k23_euclid(1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_2_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24__topreal1,t61_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e1_24_2_1_1__topreal1),[file(topreal1,e1_24_2_1_1__topreal1)]]). fof(e2_24_2_1_1__topreal1,plain,( k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),0),k4_real_1(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),k22_euclid(k23_euclid(0,1)))),k18_euclid(k21_euclid(c2_24_2_1__topreal1),2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(0,k5_real_1(1,k21_euclid(c2_24_2_1__topreal1))),k23_euclid(k4_real_1(k21_euclid(c2_24_2_1__topreal1),1),k21_euclid(c2_24_2_1__topreal1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_2_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24__topreal1,e2_24__topreal1,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e2_24_2_1_1__topreal1),[file(topreal1,e2_24_2_1_1__topreal1)]]). fof(e3_24_2_1_1__topreal1,plain,( k17_euclid(2,k23_euclid(0,k5_real_1(1,k21_euclid(c2_24_2_1__topreal1))),k23_euclid(k4_real_1(k21_euclid(c2_24_2_1__topreal1),1),k21_euclid(c2_24_2_1__topreal1))) = k23_euclid(k3_real_1(0,k21_euclid(c2_24_2_1__topreal1)),k3_real_1(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),k21_euclid(c2_24_2_1__topreal1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k17_euclid,dt_k21_euclid,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_c2_24_2_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t60_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0]), [interesting(0.35),file(topreal1,e3_24_2_1_1__topreal1),[file(topreal1,e3_24_2_1_1__topreal1)]]). fof(e4_24_2_1__topreal1,plain, ( r1_xreal_0(k21_euclid(c2_24_2_1__topreal1),1) & r1_xreal_0(0,k21_euclid(c2_24_2_1__topreal1)) & k22_euclid(c2_24_2_1__topreal1) = 1 ), inference(consider,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[dh_c2_24_2_1__topreal1,e2_24_2_1__topreal1]), [interesting(0.5),file(topreal1,e4_24_2_1__topreal1),[file(topreal1,e4_24_2_1__topreal1)]]). fof(e4_24_2_1_1__topreal1,plain,( k23_euclid(k3_real_1(0,k21_euclid(c2_24_2_1__topreal1)),k3_real_1(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),k21_euclid(c2_24_2_1__topreal1))) = c2_24_2_1__topreal1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc3_nat_1,fc3_pcomps_1,fc4_int_1,fc4_nat_1,fc4_pcomps_1,fc5_membered,fc6_int_1,fc6_membered,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_xreal_0,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k3_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_2_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e4_24_2_1__topreal1,t57_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__r1_r0_r1]), [interesting(0.35),file(topreal1,e4_24_2_1_1__topreal1),[file(topreal1,e4_24_2_1_1__topreal1)]]). fof(e5_24_2_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k21_euclid(c2_24_2_1__topreal1)),2,k23_euclid(0,1)),k18_euclid(k21_euclid(c2_24_2_1__topreal1),2,k23_euclid(1,1))) = c2_24_2_1__topreal1 ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[e1_24_2_1_1__topreal1,e2_24_2_1_1__topreal1,e3_24_2_1_1__topreal1,e4_24_2_1_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_2_1__topreal1),[file(topreal1,e5_24_2_1__topreal1)]]). fof(e3_24_2_1__topreal1,plain,( c1_24_2_1__topreal1 = c2_24_2_1__topreal1 ), inference(consider,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[dh_c2_24_2_1__topreal1,e2_24_2_1__topreal1]), [interesting(0.5),file(topreal1,e3_24_2_1__topreal1),[file(topreal1,e3_24_2_1__topreal1)]]). fof(e6_24_2_1__topreal1,plain,( r2_hidden(c1_24_2_1__topreal1,a_0_11_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_2_1__topreal1,dt_c2_24_2_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_11_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_2_1__topreal1,e3_24_2_1__topreal1,e4_24_2_1__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.5),file(topreal1,e6_24_2_1__topreal1),[file(topreal1,e6_24_2_1__topreal1)]]). fof(e7_24_2_1__topreal1,plain,( r2_hidden(c1_24_2_1__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_2_1__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_11_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_2_1__topreal1]), [interesting(0.5),file(topreal1,e7_24_2_1__topreal1),[file(topreal1,e7_24_2_1__topreal1)]]). fof(i3_24_2_1__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_2_1__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_2_1__topreal1)]). fof(i2_24_2_1__topreal1,plain,( r2_hidden(c1_24_2_1__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_2_1__topreal1,e1_24_2_1__topreal1])],[e7_24_2_1__topreal1,i3_24_2_1__topreal1]), [interesting(0.5),file(topreal1,i2_24_2_1__topreal1),[file(topreal1,i2_24_2_1__topreal1)]]). fof(i1_24_2_1__topreal1,plain,( ~ ( r2_hidden(c1_24_2_1__topreal1,a_0_1_topreal1) & ~ r2_hidden(c1_24_2_1__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_2_1__topreal1]),discharge_asm(discharge,[e1_24_2_1__topreal1])],[e1_24_2_1__topreal1,i2_24_2_1__topreal1]), [interesting(0.5),file(topreal1,i1_24_2_1__topreal1),[file(topreal1,i1_24_2_1__topreal1)]]). fof(i1_24_2_1_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_2_1__topreal1,a_0_1_topreal1) & ~ r2_hidden(c1_24_2_1__topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_2_1__topreal1])],[dt_c1_24_2_1__topreal1,i1_24_2_1__topreal1]), [interesting(0.65),e1_24_2__topreal1]). fof(e1_24_2__topreal1,plain,( r1_tarski(a_0_1_topreal1,k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1))) ), inference(let,[status(thm),assumptions([])],[i1_24_2_1_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_1_topreal1,d3_tarski,dh_c1_24_2_1__topreal1]), [interesting(0.65),file(topreal1,e1_24_2__topreal1),[file(topreal1,e1_24_2__topreal1)]]). fof(e3_24_2__topreal1,plain,( a_0_1_topreal1 = k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1)) ), inference(mizar_by,[status(thm),assumptions([])],[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_k6_xcmplx_0,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc20_xreal_0,fc4_int_1,fc4_subset_1,fc5_xreal_0,fc8_int_1,fc9_int_1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t4_arithm,free_g1_metric_1,free_g1_pre_topc,commutativity_k17_euclid,abstractness_v1_metric_1,existence_l1_metric_1,redefinition_k5_real_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_euclid,dt_k5_real_1,dt_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t1_subset,t2_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,reflexivity_r1_tarski,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,t3_subset,t2_tarski,fraenkel_a_0_1_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_2__topreal1,e1_24_2__topreal1,d10_xboole_0]), [interesting(0.65),file(topreal1,e3_24_2__topreal1),[file(topreal1,e3_24_2__topreal1)]]). fof(i1_24_2__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i1_24_2__topreal1)]), [interesting(0.65),trivial,file(topreal1,i1_24_2__topreal1)]). fof(e5_24__topreal1,plain,( a_0_1_topreal1 = k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1)) ), inference(conclusion,[status(thm),assumptions([])],[e3_24_2__topreal1,i1_24_2__topreal1]), [interesting(0.8),file(topreal1,e5_24__topreal1),[file(topreal1,e5_24__topreal1)]]). fof(fraenkel_a_0_2_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_2_topreal1) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) & A = B & r1_xreal_0(k21_euclid(B),1) & r1_xreal_0(0,k21_euclid(B)) & k22_euclid(B) = 0 ) ) ), file(topreal1,a_0_2_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_2_topreal1)]). fof(dt_c1_24_3_2__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_3_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_3_2__topreal1)]). fof(dh_c1_24_3_2__topreal1,definition, ( ~ ( r2_hidden(c1_24_3_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) & ~ r2_hidden(c1_24_3_2__topreal1,a_0_2_topreal1) ) => ! [A] : ~ ( r2_hidden(A,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) & ~ r2_hidden(A,a_0_2_topreal1) ) ), introduced(definition,[new_symbol(c1_24_3_2__topreal1),file(topreal1,c1_24_3_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_3_2__topreal1)]). fof(e1_24_3_2__topreal1,assumption,( r2_hidden(c1_24_3_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ), introduced(assumption,[file(topreal1,e1_24_3_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_3_2__topreal1)]). fof(dh_c2_24_3_2__topreal1,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_3_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(0,0)),k18_euclid(A,2,k23_euclid(1,0))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) => ( m1_subset_1(c2_24_3_2__topreal1,k1_numbers) & c1_24_3_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))) & r1_xreal_0(0,c2_24_3_2__topreal1) & r1_xreal_0(c2_24_3_2__topreal1,1) ) ), introduced(definition,[new_symbol(c2_24_3_2__topreal1),file(topreal1,c2_24_3_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_3_2__topreal1)]). fof(fraenkel_a_0_12_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_12_topreal1) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & A = k17_euclid(2,k18_euclid(k5_real_1(1,B),2,k23_euclid(0,0)),k18_euclid(B,2,k23_euclid(1,0))) & r1_xreal_0(0,B) & r1_xreal_0(B,1) ) ) ), file(topreal1,a_0_12_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_12_topreal1)]). fof(e2_24_3_2__topreal1,plain,( r2_hidden(c1_24_3_2__topreal1,a_0_12_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_3_2__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_12_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_3_2__topreal1]), [interesting(0.5),file(topreal1,e2_24_3_2__topreal1),[file(topreal1,e2_24_3_2__topreal1)]]). fof(e3_24_3_2__topreal1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_3_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(0,0)),k18_euclid(A,2,k23_euclid(1,0))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_arithm,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_topreal1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k5_real_1,dt_m1_subset_1,dt_c1_24_3_2__topreal1,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_12_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_3_2__topreal1]), [interesting(0.5),file(topreal1,e3_24_3_2__topreal1),[file(topreal1,e3_24_3_2__topreal1)]]). fof(dt_c2_24_3_2__topreal1,plain,( m1_subset_1(c2_24_3_2__topreal1,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[dh_c2_24_3_2__topreal1,e3_24_3_2__topreal1]), [interesting(0.5),file(topreal1,c2_24_3_2__topreal1),[file(topreal1,c2_24_3_2__topreal1)]]). fof(e1_24_3_2_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))) = k17_euclid(2,k16_euclid(2),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[existence_m1_finseq_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_finseq_2,dt_k2_zfmisc_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k4_finseq_2,dt_k4_finseqop,dt_l1_metric_1,dt_m1_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k14_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,d7_euclid,d1_euclid,d4_euclid,existence_m1_subset_1,redefinition_k5_euclid,dt_k10_finseq_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_subset_1,fc2_euclid,fc2_topreal1,fc5_int_1,fc5_membered,fc5_xreal_0,fc9_int_1,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_subset_1,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_m2_subset_1,dt_c2_24_3_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc2_membered,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t32_euclid,t58_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e1_24_3_2_1__topreal1),[file(topreal1,e1_24_3_2_1__topreal1)]]). fof(e2_24_3_2_1__topreal1,plain,( k17_euclid(2,k16_euclid(2),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))) = k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k4_finseqop,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_l1_metric_1,dt_m2_finseq_2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_membered,rc1_metric_1,rc2_metric_1,rc3_metric_1,t1_subset,t4_subset,t5_subset,d1_euclid,d4_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k5_euclid,dt_k10_finseq_1,dt_k14_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc5_membered,rc1_int_1,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c2_24_3_2__topreal1,cc1_nat_1,cc2_int_1,cc2_nat_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_numerals,d8_euclid,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t31_euclid]), [interesting(0.35),file(topreal1,e2_24_3_2_1__topreal1),[file(topreal1,e2_24_3_2_1__topreal1)]]). fof(e3_24_3_2_1__topreal1,plain,( k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0)) = k23_euclid(k4_real_1(c2_24_3_2__topreal1,k21_euclid(k23_euclid(1,0))),k4_real_1(c2_24_3_2__topreal1,k22_euclid(k23_euclid(1,0)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m1_subset_1,redefinition_k4_real_1,dt_k15_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_3_2__topreal1,cc2_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e3_24_3_2_1__topreal1),[file(topreal1,e3_24_3_2_1__topreal1)]]). fof(e3_24__topreal1,plain, ( k21_euclid(k23_euclid(1,0)) = 1 & k22_euclid(k23_euclid(1,0)) = 0 ), inference(mizar_by,[status(thm),assumptions([])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc2_boole,spc2_numerals,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,cc2_xreal_0,d16_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t56_euclid]), [interesting(0.8),file(topreal1,e3_24__topreal1),[file(topreal1,e3_24__topreal1)]]). fof(e4_24_3_2_1__topreal1,plain,( k23_euclid(k4_real_1(c2_24_3_2__topreal1,k21_euclid(k23_euclid(1,0))),k4_real_1(c2_24_3_2__topreal1,k22_euclid(k23_euclid(1,0)))) = k23_euclid(c2_24_3_2__topreal1,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc4_xreal_0,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,spc2_boole,spc2_numerals,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,commutativity_k3_xcmplx_0,commutativity_k4_real_1,redefinition_k4_real_1,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_c2_24_3_2__topreal1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,d16_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e3_24__topreal1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e4_24_3_2_1__topreal1),[file(topreal1,e4_24_3_2_1__topreal1)]]). fof(e5_24_3_2__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))) = k23_euclid(c2_24_3_2__topreal1,0) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[e1_24_3_2_1__topreal1,e2_24_3_2_1__topreal1,e3_24_3_2_1__topreal1,e4_24_3_2_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_3_2__topreal1),[file(topreal1,e5_24_3_2__topreal1)]]). fof(e4_24_3_2__topreal1,plain, ( c1_24_3_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))) & r1_xreal_0(0,c2_24_3_2__topreal1) & r1_xreal_0(c2_24_3_2__topreal1,1) ), inference(consider,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[dh_c2_24_3_2__topreal1,e3_24_3_2__topreal1]), [interesting(0.5),file(topreal1,e4_24_3_2__topreal1),[file(topreal1,e4_24_3_2__topreal1)]]). fof(e6_24_3_2__topreal1,plain, ( r1_xreal_0(k21_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0)))),1) & r1_xreal_0(0,k21_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0))))) & k22_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_3_2__topreal1),2,k23_euclid(0,0)),k18_euclid(c2_24_3_2__topreal1,2,k23_euclid(1,0)))) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_3_2__topreal1,dt_c2_24_3_2__topreal1,cc2_xreal_0,fc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_3_2__topreal1,e4_24_3_2__topreal1,t56_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.5),file(topreal1,e6_24_3_2__topreal1),[file(topreal1,e6_24_3_2__topreal1)]]). fof(e7_24_3_2__topreal1,plain,( r2_hidden(c1_24_3_2__topreal1,a_0_2_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_3_2__topreal1,dt_c2_24_3_2__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_2_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_3_2__topreal1,e4_24_3_2__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2]), [interesting(0.5),file(topreal1,e7_24_3_2__topreal1),[file(topreal1,e7_24_3_2__topreal1)]]). fof(i3_24_3_2__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_3_2__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_3_2__topreal1)]). fof(i2_24_3_2__topreal1,plain,( r2_hidden(c1_24_3_2__topreal1,a_0_2_topreal1) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_3_2__topreal1,e1_24_3_2__topreal1])],[e7_24_3_2__topreal1,i3_24_3_2__topreal1]), [interesting(0.5),file(topreal1,i2_24_3_2__topreal1),[file(topreal1,i2_24_3_2__topreal1)]]). fof(i1_24_3_2__topreal1,plain,( ~ ( r2_hidden(c1_24_3_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) & ~ r2_hidden(c1_24_3_2__topreal1,a_0_2_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_3_2__topreal1]),discharge_asm(discharge,[e1_24_3_2__topreal1])],[e1_24_3_2__topreal1,i2_24_3_2__topreal1]), [interesting(0.5),file(topreal1,i1_24_3_2__topreal1),[file(topreal1,i1_24_3_2__topreal1)]]). fof(i1_24_3_2_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_3_2__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) & ~ r2_hidden(c1_24_3_2__topreal1,a_0_2_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_3_2__topreal1])],[dt_c1_24_3_2__topreal1,i1_24_3_2__topreal1]), [interesting(0.65),e2_24_3__topreal1]). fof(e2_24_3__topreal1,plain,( r1_tarski(k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0)),a_0_2_topreal1) ), inference(let,[status(thm),assumptions([])],[i1_24_3_2_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_2_topreal1,d3_tarski,dh_c1_24_3_2__topreal1]), [interesting(0.65),file(topreal1,e2_24_3__topreal1),[file(topreal1,e2_24_3__topreal1)]]). fof(dt_c1_24_3_1__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_3_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_3_1__topreal1)]). fof(dh_c1_24_3_1__topreal1,definition, ( ~ ( r2_hidden(c1_24_3_1__topreal1,a_0_2_topreal1) & ~ r2_hidden(c1_24_3_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ) => ! [A] : ~ ( r2_hidden(A,a_0_2_topreal1) & ~ r2_hidden(A,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ) ), introduced(definition,[new_symbol(c1_24_3_1__topreal1),file(topreal1,c1_24_3_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_3_1__topreal1)]). fof(e1_24_3_1__topreal1,assumption,( r2_hidden(c1_24_3_1__topreal1,a_0_2_topreal1) ), introduced(assumption,[file(topreal1,e1_24_3_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_3_1__topreal1)]). fof(dh_c2_24_3_1__topreal1,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_3_1__topreal1 = A & r1_xreal_0(k21_euclid(A),1) & r1_xreal_0(0,k21_euclid(A)) & k22_euclid(A) = 0 ) => ( m1_subset_1(c2_24_3_1__topreal1,u1_struct_0(k15_euclid(2))) & c1_24_3_1__topreal1 = c2_24_3_1__topreal1 & r1_xreal_0(k21_euclid(c2_24_3_1__topreal1),1) & r1_xreal_0(0,k21_euclid(c2_24_3_1__topreal1)) & k22_euclid(c2_24_3_1__topreal1) = 0 ) ), introduced(definition,[new_symbol(c2_24_3_1__topreal1),file(topreal1,c2_24_3_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_3_1__topreal1)]). fof(e2_24_3_1__topreal1,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_3_1__topreal1 = A & r1_xreal_0(k21_euclid(A),1) & r1_xreal_0(0,k21_euclid(A)) & k22_euclid(A) = 0 ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_1,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_k1_xboole_0,dt_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_m1_subset_1,dt_u1_struct_0,dt_c1_24_3_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_2_topreal1,d8_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_3_1__topreal1]), [interesting(0.5),file(topreal1,e2_24_3_1__topreal1),[file(topreal1,e2_24_3_1__topreal1)]]). fof(dt_c2_24_3_1__topreal1,plain,( m1_subset_1(c2_24_3_1__topreal1,u1_struct_0(k15_euclid(2))) ), inference(consider,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[dh_c2_24_3_1__topreal1,e2_24_3_1__topreal1]), [interesting(0.5),file(topreal1,c2_24_3_1__topreal1),[file(topreal1,c2_24_3_1__topreal1)]]). fof(e1_24_3_1_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k21_euclid(c2_24_3_1__topreal1)),2,k23_euclid(0,0)),k18_euclid(k21_euclid(c2_24_3_1__topreal1),2,k23_euclid(1,0))) = k17_euclid(2,k16_euclid(2),k18_euclid(k21_euclid(c2_24_3_1__topreal1),2,k23_euclid(1,0))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[existence_m1_finseq_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k2_finseq_2,dt_k2_zfmisc_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,abstractness_v1_metric_1,existence_l1_metric_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k4_finseq_2,dt_k4_finseqop,dt_l1_metric_1,dt_m1_finseq_2,dt_m2_finseq_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k14_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t4_subset,t5_subset,d7_euclid,d1_euclid,d4_euclid,existence_m1_subset_1,redefinition_k5_euclid,dt_k10_finseq_1,dt_k15_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_m1_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_subset_1,fc2_euclid,fc2_topreal1,fc5_int_1,fc5_membered,fc5_xreal_0,fc9_int_1,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_subset_1,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k21_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_m2_subset_1,dt_c2_24_3_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc2_membered,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t32_euclid,t58_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e1_24_3_1_1__topreal1),[file(topreal1,e1_24_3_1_1__topreal1)]]). fof(e2_24_3_1_1__topreal1,plain,( k17_euclid(2,k16_euclid(2),k18_euclid(k21_euclid(c2_24_3_1__topreal1),2,k23_euclid(1,0))) = k18_euclid(k21_euclid(c2_24_3_1__topreal1),2,k23_euclid(1,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[existence_m1_finseq_1,dt_k2_finseq_2,dt_m1_finseq_1,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_k4_finseqop,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k2_zfmisc_1,dt_k4_finseq_2,dt_k4_finseqop,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc4_subset_1,free_g1_metric_1,free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,abstractness_v1_metric_1,existence_l1_metric_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k1_euclid,dt_k1_xboole_0,dt_k4_euclid,dt_l1_metric_1,dt_m2_finseq_2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc6_membered,rc1_membered,rc1_metric_1,rc2_metric_1,rc3_metric_1,t1_subset,t4_subset,t5_subset,d1_euclid,d4_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,redefinition_k5_euclid,dt_k10_finseq_1,dt_k14_euclid,dt_k1_zfmisc_1,dt_k5_euclid,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_euclid,fc1_struct_0,fc1_subset_1,fc5_membered,rc1_int_1,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xboole_0,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xboole_0,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k16_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k21_euclid,dt_k23_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c2_24_3_1__topreal1,cc1_nat_1,cc2_int_1,cc2_nat_1,fc2_euclid,fc2_membered,fc2_topreal1,t1_numerals,d8_euclid,d9_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t31_euclid]), [interesting(0.35),file(topreal1,e2_24_3_1_1__topreal1),[file(topreal1,e2_24_3_1_1__topreal1)]]). fof(e3_24_3_1_1__topreal1,plain,( k18_euclid(k21_euclid(c2_24_3_1__topreal1),2,k23_euclid(1,0)) = k23_euclid(k4_real_1(k21_euclid(c2_24_3_1__topreal1),k21_euclid(k23_euclid(1,0))),k4_real_1(k21_euclid(c2_24_3_1__topreal1),k22_euclid(k23_euclid(1,0)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc7_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,existence_m1_subset_1,redefinition_k4_real_1,dt_k15_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_3_1__topreal1,cc2_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e3_24_3_1_1__topreal1),[file(topreal1,e3_24_3_1_1__topreal1)]]). fof(e4_24_3_1__topreal1,plain, ( r1_xreal_0(k21_euclid(c2_24_3_1__topreal1),1) & r1_xreal_0(0,k21_euclid(c2_24_3_1__topreal1)) & k22_euclid(c2_24_3_1__topreal1) = 0 ), inference(consider,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[dh_c2_24_3_1__topreal1,e2_24_3_1__topreal1]), [interesting(0.5),file(topreal1,e4_24_3_1__topreal1),[file(topreal1,e4_24_3_1__topreal1)]]). fof(e4_24_3_1_1__topreal1,plain,( k23_euclid(k4_real_1(k21_euclid(c2_24_3_1__topreal1),k21_euclid(k23_euclid(1,0))),k4_real_1(k21_euclid(c2_24_3_1__topreal1),k22_euclid(k23_euclid(1,0)))) = c2_24_3_1__topreal1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc4_xreal_0,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,spc7_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k4_real_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_3_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_24__topreal1,e4_24_3_1__topreal1,t57_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e4_24_3_1_1__topreal1),[file(topreal1,e4_24_3_1_1__topreal1)]]). fof(e5_24_3_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k21_euclid(c2_24_3_1__topreal1)),2,k23_euclid(0,0)),k18_euclid(k21_euclid(c2_24_3_1__topreal1),2,k23_euclid(1,0))) = c2_24_3_1__topreal1 ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[e1_24_3_1_1__topreal1,e2_24_3_1_1__topreal1,e3_24_3_1_1__topreal1,e4_24_3_1_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_3_1__topreal1),[file(topreal1,e5_24_3_1__topreal1)]]). fof(e3_24_3_1__topreal1,plain,( c1_24_3_1__topreal1 = c2_24_3_1__topreal1 ), inference(consider,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[dh_c2_24_3_1__topreal1,e2_24_3_1__topreal1]), [interesting(0.5),file(topreal1,e3_24_3_1__topreal1),[file(topreal1,e3_24_3_1__topreal1)]]). fof(e6_24_3_1__topreal1,plain,( r2_hidden(c1_24_3_1__topreal1,a_0_12_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_3_1__topreal1,dt_c2_24_3_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_12_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_3_1__topreal1,e3_24_3_1__topreal1,e4_24_3_1__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.5),file(topreal1,e6_24_3_1__topreal1),[file(topreal1,e6_24_3_1__topreal1)]]). fof(e7_24_3_1__topreal1,plain,( r2_hidden(c1_24_3_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_3_1__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_12_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_3_1__topreal1]), [interesting(0.5),file(topreal1,e7_24_3_1__topreal1),[file(topreal1,e7_24_3_1__topreal1)]]). fof(i3_24_3_1__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_3_1__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_3_1__topreal1)]). fof(i2_24_3_1__topreal1,plain,( r2_hidden(c1_24_3_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_3_1__topreal1,e1_24_3_1__topreal1])],[e7_24_3_1__topreal1,i3_24_3_1__topreal1]), [interesting(0.5),file(topreal1,i2_24_3_1__topreal1),[file(topreal1,i2_24_3_1__topreal1)]]). fof(i1_24_3_1__topreal1,plain,( ~ ( r2_hidden(c1_24_3_1__topreal1,a_0_2_topreal1) & ~ r2_hidden(c1_24_3_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_3_1__topreal1]),discharge_asm(discharge,[e1_24_3_1__topreal1])],[e1_24_3_1__topreal1,i2_24_3_1__topreal1]), [interesting(0.5),file(topreal1,i1_24_3_1__topreal1),[file(topreal1,i1_24_3_1__topreal1)]]). fof(i1_24_3_1_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_3_1__topreal1,a_0_2_topreal1) & ~ r2_hidden(c1_24_3_1__topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_3_1__topreal1])],[dt_c1_24_3_1__topreal1,i1_24_3_1__topreal1]), [interesting(0.65),e1_24_3__topreal1]). fof(e1_24_3__topreal1,plain,( r1_tarski(a_0_2_topreal1,k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0))) ), inference(let,[status(thm),assumptions([])],[i1_24_3_1_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_2_topreal1,d3_tarski,dh_c1_24_3_1__topreal1]), [interesting(0.65),file(topreal1,e1_24_3__topreal1),[file(topreal1,e1_24_3__topreal1)]]). fof(e3_24_3__topreal1,plain,( a_0_2_topreal1 = k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0)) ), inference(mizar_by,[status(thm),assumptions([])],[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_k6_xcmplx_0,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc20_xreal_0,fc4_int_1,fc4_subset_1,fc5_xreal_0,fc8_int_1,fc9_int_1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t4_arithm,free_g1_metric_1,free_g1_pre_topc,commutativity_k17_euclid,abstractness_v1_metric_1,existence_l1_metric_1,redefinition_k5_real_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_euclid,dt_k5_real_1,dt_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t1_subset,t2_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,reflexivity_r1_tarski,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,t3_subset,t2_tarski,fraenkel_a_0_2_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_3__topreal1,e1_24_3__topreal1,d10_xboole_0]), [interesting(0.65),file(topreal1,e3_24_3__topreal1),[file(topreal1,e3_24_3__topreal1)]]). fof(i1_24_3__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i1_24_3__topreal1)]), [interesting(0.65),trivial,file(topreal1,i1_24_3__topreal1)]). fof(e6_24__topreal1,plain,( a_0_2_topreal1 = k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0)) ), inference(conclusion,[status(thm),assumptions([])],[e3_24_3__topreal1,i1_24_3__topreal1]), [interesting(0.8),file(topreal1,e6_24__topreal1),[file(topreal1,e6_24__topreal1)]]). fof(fraenkel_a_0_3_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_3_topreal1) <=> ? [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(2))) & A = B & k21_euclid(B) = 1 & r1_xreal_0(k22_euclid(B),1) & r1_xreal_0(0,k22_euclid(B)) ) ) ), file(topreal1,a_0_3_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_3_topreal1)]). fof(dt_c1_24_4_2__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_4_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_4_2__topreal1)]). fof(dh_c1_24_4_2__topreal1,definition, ( ~ ( r2_hidden(c1_24_4_2__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) & ~ r2_hidden(c1_24_4_2__topreal1,a_0_3_topreal1) ) => ! [A] : ~ ( r2_hidden(A,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) & ~ r2_hidden(A,a_0_3_topreal1) ) ), introduced(definition,[new_symbol(c1_24_4_2__topreal1),file(topreal1,c1_24_4_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_4_2__topreal1)]). fof(e1_24_4_2__topreal1,assumption,( r2_hidden(c1_24_4_2__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ), introduced(assumption,[file(topreal1,e1_24_4_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_4_2__topreal1)]). fof(dh_c2_24_4_2__topreal1,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_4_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(1,0)),k18_euclid(A,2,k23_euclid(1,1))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) => ( m1_subset_1(c2_24_4_2__topreal1,k1_numbers) & c1_24_4_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))) & r1_xreal_0(0,c2_24_4_2__topreal1) & r1_xreal_0(c2_24_4_2__topreal1,1) ) ), introduced(definition,[new_symbol(c2_24_4_2__topreal1),file(topreal1,c2_24_4_2__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_4_2__topreal1)]). fof(fraenkel_a_0_13_topreal1,definition,( ! [A] : ( r2_hidden(A,a_0_13_topreal1) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & A = k17_euclid(2,k18_euclid(k5_real_1(1,B),2,k23_euclid(1,0)),k18_euclid(B,2,k23_euclid(1,1))) & r1_xreal_0(0,B) & r1_xreal_0(B,1) ) ) ), file(topreal1,a_0_13_topreal1), [interesting(0.9),axiom,file(topreal1,a_0_13_topreal1)]). fof(e2_24_4_2__topreal1,plain,( r2_hidden(c1_24_4_2__topreal1,a_0_13_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_4_2__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_13_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_4_2__topreal1]), [interesting(0.5),file(topreal1,e2_24_4_2__topreal1),[file(topreal1,e2_24_4_2__topreal1)]]). fof(e3_24_4_2__topreal1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & c1_24_4_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,A),2,k23_euclid(1,0)),k18_euclid(A,2,k23_euclid(1,1))) & r1_xreal_0(0,A) & r1_xreal_0(A,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_arithm,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_topreal1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k23_euclid,dt_k5_real_1,dt_m1_subset_1,dt_c1_24_4_2__topreal1,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_13_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_4_2__topreal1]), [interesting(0.5),file(topreal1,e3_24_4_2__topreal1),[file(topreal1,e3_24_4_2__topreal1)]]). fof(dt_c2_24_4_2__topreal1,plain,( m1_subset_1(c2_24_4_2__topreal1,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[dh_c2_24_4_2__topreal1,e3_24_4_2__topreal1]), [interesting(0.5),file(topreal1,c2_24_4_2__topreal1),[file(topreal1,c2_24_4_2__topreal1)]]). fof(e1_24_4_2_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,c2_24_4_2__topreal1),1),k4_real_1(k5_real_1(1,c2_24_4_2__topreal1),k22_euclid(k23_euclid(1,0)))),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_4_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_24__topreal1,t61_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e1_24_4_2_1__topreal1),[file(topreal1,e1_24_4_2_1__topreal1)]]). fof(e2_24_4_2_1__topreal1,plain,( k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,c2_24_4_2__topreal1),1),k4_real_1(k5_real_1(1,c2_24_4_2__topreal1),k22_euclid(k23_euclid(1,0)))),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(k5_real_1(1,c2_24_4_2__topreal1),0),k23_euclid(c2_24_4_2__topreal1,k4_real_1(c2_24_4_2__topreal1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_4_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24__topreal1,e3_24__topreal1,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e2_24_4_2_1__topreal1),[file(topreal1,e2_24_4_2_1__topreal1)]]). fof(e3_24_4_2_1__topreal1,plain,( k17_euclid(2,k23_euclid(k5_real_1(1,c2_24_4_2__topreal1),0),k23_euclid(c2_24_4_2__topreal1,k4_real_1(c2_24_4_2__topreal1,1))) = k23_euclid(k3_real_1(k5_real_1(1,c2_24_4_2__topreal1),c2_24_4_2__topreal1),k3_real_1(0,c2_24_4_2__topreal1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k17_euclid,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_c2_24_4_2__topreal1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t60_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0]), [interesting(0.35),file(topreal1,e3_24_4_2_1__topreal1),[file(topreal1,e3_24_4_2_1__topreal1)]]). fof(e4_24_4_2_1__topreal1,plain,( k23_euclid(k3_real_1(k5_real_1(1,c2_24_4_2__topreal1),c2_24_4_2__topreal1),k3_real_1(0,c2_24_4_2__topreal1)) = k23_euclid(1,c2_24_4_2__topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_xreal_0,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc1_arithm,spc2_boole,spc2_numerals,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,spc2_numerals,spc2_boole,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k5_real_1,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_c2_24_4_2__topreal1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,d16_euclid,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__r1_r0_r1]), [interesting(0.35),file(topreal1,e4_24_4_2_1__topreal1),[file(topreal1,e4_24_4_2_1__topreal1)]]). fof(e5_24_4_2__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))) = k23_euclid(1,c2_24_4_2__topreal1) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[e1_24_4_2_1__topreal1,e2_24_4_2_1__topreal1,e3_24_4_2_1__topreal1,e4_24_4_2_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_4_2__topreal1),[file(topreal1,e5_24_4_2__topreal1)]]). fof(e4_24_4_2__topreal1,plain, ( c1_24_4_2__topreal1 = k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))) & r1_xreal_0(0,c2_24_4_2__topreal1) & r1_xreal_0(c2_24_4_2__topreal1,1) ), inference(consider,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[dh_c2_24_4_2__topreal1,e3_24_4_2__topreal1]), [interesting(0.5),file(topreal1,e4_24_4_2__topreal1),[file(topreal1,e4_24_4_2__topreal1)]]). fof(e6_24_4_2__topreal1,plain, ( k21_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1)))) = 1 & r1_xreal_0(k22_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1)))),1) & r1_xreal_0(0,k22_euclid(k17_euclid(2,k18_euclid(k5_real_1(1,c2_24_4_2__topreal1),2,k23_euclid(1,0)),k18_euclid(c2_24_4_2__topreal1,2,k23_euclid(1,1))))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_4_2__topreal1,dt_c2_24_4_2__topreal1,cc2_xreal_0,fc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_4_2__topreal1,e4_24_4_2__topreal1,t56_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.5),file(topreal1,e6_24_4_2__topreal1),[file(topreal1,e6_24_4_2__topreal1)]]). fof(e7_24_4_2__topreal1,plain,( r2_hidden(c1_24_4_2__topreal1,a_0_3_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_4_2__topreal1,dt_c2_24_4_2__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_3_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_4_2__topreal1,e4_24_4_2__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2]), [interesting(0.5),file(topreal1,e7_24_4_2__topreal1),[file(topreal1,e7_24_4_2__topreal1)]]). fof(i3_24_4_2__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_4_2__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_4_2__topreal1)]). fof(i2_24_4_2__topreal1,plain,( r2_hidden(c1_24_4_2__topreal1,a_0_3_topreal1) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_4_2__topreal1,e1_24_4_2__topreal1])],[e7_24_4_2__topreal1,i3_24_4_2__topreal1]), [interesting(0.5),file(topreal1,i2_24_4_2__topreal1),[file(topreal1,i2_24_4_2__topreal1)]]). fof(i1_24_4_2__topreal1,plain,( ~ ( r2_hidden(c1_24_4_2__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) & ~ r2_hidden(c1_24_4_2__topreal1,a_0_3_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_4_2__topreal1]),discharge_asm(discharge,[e1_24_4_2__topreal1])],[e1_24_4_2__topreal1,i2_24_4_2__topreal1]), [interesting(0.5),file(topreal1,i1_24_4_2__topreal1),[file(topreal1,i1_24_4_2__topreal1)]]). fof(i1_24_4_2_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_4_2__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) & ~ r2_hidden(c1_24_4_2__topreal1,a_0_3_topreal1) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_4_2__topreal1])],[dt_c1_24_4_2__topreal1,i1_24_4_2__topreal1]), [interesting(0.65),e2_24_4__topreal1]). fof(e2_24_4__topreal1,plain,( r1_tarski(k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)),a_0_3_topreal1) ), inference(let,[status(thm),assumptions([])],[i1_24_4_2_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_3_topreal1,d3_tarski,dh_c1_24_4_2__topreal1]), [interesting(0.65),file(topreal1,e2_24_4__topreal1),[file(topreal1,e2_24_4__topreal1)]]). fof(dt_c1_24_4_1__topreal1,assumption,( $true ), introduced(assumption,[file(topreal1,c1_24_4_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_4_1__topreal1)]). fof(dh_c1_24_4_1__topreal1,definition, ( ~ ( r2_hidden(c1_24_4_1__topreal1,a_0_3_topreal1) & ~ r2_hidden(c1_24_4_1__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ) => ! [A] : ~ ( r2_hidden(A,a_0_3_topreal1) & ~ r2_hidden(A,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ) ), introduced(definition,[new_symbol(c1_24_4_1__topreal1),file(topreal1,c1_24_4_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c1_24_4_1__topreal1)]). fof(e1_24_4_1__topreal1,assumption,( r2_hidden(c1_24_4_1__topreal1,a_0_3_topreal1) ), introduced(assumption,[file(topreal1,e1_24_4_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,e1_24_4_1__topreal1)]). fof(dh_c2_24_4_1__topreal1,definition, ( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_4_1__topreal1 = A & k21_euclid(A) = 1 & r1_xreal_0(k22_euclid(A),1) & r1_xreal_0(0,k22_euclid(A)) ) => ( m1_subset_1(c2_24_4_1__topreal1,u1_struct_0(k15_euclid(2))) & c1_24_4_1__topreal1 = c2_24_4_1__topreal1 & k21_euclid(c2_24_4_1__topreal1) = 1 & r1_xreal_0(k22_euclid(c2_24_4_1__topreal1),1) & r1_xreal_0(0,k22_euclid(c2_24_4_1__topreal1)) ) ), introduced(definition,[new_symbol(c2_24_4_1__topreal1),file(topreal1,c2_24_4_1__topreal1)]), [interesting(0.5),axiom,file(topreal1,c2_24_4_1__topreal1)]). fof(e2_24_4_1__topreal1,plain,( ? [A] : ( m1_subset_1(A,u1_struct_0(k15_euclid(2))) & c1_24_4_1__topreal1 = A & k21_euclid(A) = 1 & r1_xreal_0(k22_euclid(A),1) & r1_xreal_0(0,k22_euclid(A)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_1,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_k1_xboole_0,dt_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc3_pcomps_1,fc4_pcomps_1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc1_xreal_0,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k5_numbers,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_euclid,fc2_euclid,fc2_membered,fc2_topreal1,rc1_pre_topc,rc1_xboole_0,rc2_xboole_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t8_boole,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_m1_subset_1,dt_u1_struct_0,dt_c1_24_4_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_3_topreal1,d8_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_24_4_1__topreal1]), [interesting(0.5),file(topreal1,e2_24_4_1__topreal1),[file(topreal1,e2_24_4_1__topreal1)]]). fof(dt_c2_24_4_1__topreal1,plain,( m1_subset_1(c2_24_4_1__topreal1,u1_struct_0(k15_euclid(2))) ), inference(consider,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[dh_c2_24_4_1__topreal1,e2_24_4_1__topreal1]), [interesting(0.5),file(topreal1,c2_24_4_1__topreal1),[file(topreal1,c2_24_4_1__topreal1)]]). fof(e1_24_4_1_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),2,k23_euclid(1,0)),k18_euclid(k22_euclid(c2_24_4_1__topreal1),2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),1),k4_real_1(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),k22_euclid(k23_euclid(1,0)))),k18_euclid(k22_euclid(c2_24_4_1__topreal1),2,k23_euclid(1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_4_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_24__topreal1,t61_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.35),file(topreal1,e1_24_4_1_1__topreal1),[file(topreal1,e1_24_4_1_1__topreal1)]]). fof(e2_24_4_1_1__topreal1,plain,( k17_euclid(2,k23_euclid(k4_real_1(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),1),k4_real_1(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),k22_euclid(k23_euclid(1,0)))),k18_euclid(k22_euclid(c2_24_4_1__topreal1),2,k23_euclid(1,1))) = k17_euclid(2,k23_euclid(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),0),k23_euclid(k4_real_1(k22_euclid(c2_24_4_1__topreal1),1),k22_euclid(c2_24_4_1__topreal1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_pcomps_1,fc4_int_1,fc4_pcomps_1,fc5_membered,fc6_membered,fc7_int_1,fc8_int_1,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc2_arithm,spc7_arithm,spc9_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k17_euclid,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,existence_m1_subset_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_4_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc4_xreal_0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24__topreal1,e3_24__topreal1,t61_euclid,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.35),file(topreal1,e2_24_4_1_1__topreal1),[file(topreal1,e2_24_4_1_1__topreal1)]]). fof(e3_24_4_1_1__topreal1,plain,( k17_euclid(2,k23_euclid(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),0),k23_euclid(k4_real_1(k22_euclid(c2_24_4_1__topreal1),1),k22_euclid(c2_24_4_1__topreal1))) = k23_euclid(k3_real_1(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),k22_euclid(c2_24_4_1__topreal1)),k3_real_1(0,k22_euclid(c2_24_4_1__topreal1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t3_subset,t4_subset,t5_subset,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc23_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t4_arithm,t6_boole,t7_boole,t8_boole,d8_euclid,commutativity_k17_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,redefinition_k3_real_1,redefinition_k4_real_1,redefinition_k5_real_1,dt_k17_euclid,dt_k22_euclid,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_c2_24_4_1__topreal1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r0_r2_r0,rqRealMult__k3_xcmplx_0__r0_rm2_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r2_r2,rqRealMult__k3_xcmplx_0__r1_rm2_rm2,rqRealMult__k3_xcmplx_0__r2_r0_r0,rqRealMult__k3_xcmplx_0__r2_r1_r2,rqRealMult__k3_xcmplx_0__rm2_r0_r0,rqRealMult__k3_xcmplx_0__rm2_r1_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t60_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0]), [interesting(0.35),file(topreal1,e3_24_4_1_1__topreal1),[file(topreal1,e3_24_4_1_1__topreal1)]]). fof(e4_24_4_1__topreal1,plain, ( k21_euclid(c2_24_4_1__topreal1) = 1 & r1_xreal_0(k22_euclid(c2_24_4_1__topreal1),1) & r1_xreal_0(0,k22_euclid(c2_24_4_1__topreal1)) ), inference(consider,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[dh_c2_24_4_1__topreal1,e2_24_4_1__topreal1]), [interesting(0.5),file(topreal1,e4_24_4_1__topreal1),[file(topreal1,e4_24_4_1__topreal1)]]). fof(e4_24_4_1_1__topreal1,plain,( k23_euclid(k3_real_1(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),k22_euclid(c2_24_4_1__topreal1)),k3_real_1(0,k22_euclid(c2_24_4_1__topreal1))) = c2_24_4_1__topreal1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[reflexivity_r1_tarski,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_relset_1,fc4_subset_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_k1_zfmisc_1,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,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc3_nat_1,fc3_pcomps_1,fc4_int_1,fc4_nat_1,fc4_pcomps_1,fc5_membered,fc6_int_1,fc6_membered,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_metric_1,rc1_nat_1,rc1_subset_1,rc2_int_1,rc2_metric_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_metric_1,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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_k10_finseq_1,dt_k14_euclid,dt_k1_numbers,dt_k5_numbers,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_euclid,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc3_xreal_0,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_pre_topc,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,d7_euclid,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k3_real_1,redefinition_k5_real_1,dt_k15_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_real_1,dt_m1_subset_1,dt_u1_struct_0,dt_c2_24_4_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,d8_euclid,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e4_24_4_1__topreal1,t57_euclid,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__r1_r0_r1]), [interesting(0.35),file(topreal1,e4_24_4_1_1__topreal1),[file(topreal1,e4_24_4_1_1__topreal1)]]). fof(e5_24_4_1__topreal1,plain,( k17_euclid(2,k18_euclid(k5_real_1(1,k22_euclid(c2_24_4_1__topreal1)),2,k23_euclid(1,0)),k18_euclid(k22_euclid(c2_24_4_1__topreal1),2,k23_euclid(1,1))) = c2_24_4_1__topreal1 ), inference(iterative_eq,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[e1_24_4_1_1__topreal1,e2_24_4_1_1__topreal1,e3_24_4_1_1__topreal1,e4_24_4_1_1__topreal1]), [interesting(0.5),file(topreal1,e5_24_4_1__topreal1),[file(topreal1,e5_24_4_1__topreal1)]]). fof(e3_24_4_1__topreal1,plain,( c1_24_4_1__topreal1 = c2_24_4_1__topreal1 ), inference(consider,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[dh_c2_24_4_1__topreal1,e2_24_4_1__topreal1]), [interesting(0.5),file(topreal1,e3_24_4_1__topreal1),[file(topreal1,e3_24_4_1__topreal1)]]). fof(e6_24_4_1__topreal1,plain,( r2_hidden(c1_24_4_1__topreal1,a_0_13_topreal1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[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_relset_1,fc4_subset_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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_subset_1,fc1_xboole_0,fc20_xreal_0,fc3_int_1,fc4_int_1,fc5_membered,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_subset_1,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_subset_1,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,d7_euclid,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc2_topreal1,fc5_int_1,fc5_xreal_0,fc9_int_1,rc1_xboole_0,rc1_xreal_0,rc2_xboole_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,spc9_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t8_boole,d8_euclid,commutativity_k17_euclid,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k5_real_1,dt_k17_euclid,dt_k18_euclid,dt_k21_euclid,dt_k22_euclid,dt_k23_euclid,dt_k4_xcmplx_0,dt_k5_real_1,dt_c1_24_4_1__topreal1,dt_c2_24_4_1__topreal1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_13_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e5_24_4_1__topreal1,e3_24_4_1__topreal1,e4_24_4_1__topreal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r2]), [interesting(0.5),file(topreal1,e6_24_4_1__topreal1),[file(topreal1,e6_24_4_1__topreal1)]]). fof(e7_24_4_1__topreal1,plain,( r2_hidden(c1_24_4_1__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[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_relset_1,fc4_subset_1,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_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,reflexivity_r1_tarski,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_k6_xcmplx_0,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc20_xreal_0,fc4_int_1,fc5_membered,fc5_xreal_0,fc6_membered,fc8_int_1,fc9_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t2_real,t3_real,t4_arithm,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,commutativity_k17_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k5_numbers,dt_k5_real_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,dt_c1_24_4_1__topreal1,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_13_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_24_4_1__topreal1]), [interesting(0.5),file(topreal1,e7_24_4_1__topreal1),[file(topreal1,e7_24_4_1__topreal1)]]). fof(i3_24_4_1__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i3_24_4_1__topreal1)]), [interesting(0.5),trivial,file(topreal1,i3_24_4_1__topreal1)]). fof(i2_24_4_1__topreal1,plain,( r2_hidden(c1_24_4_1__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_24_4_1__topreal1,e1_24_4_1__topreal1])],[e7_24_4_1__topreal1,i3_24_4_1__topreal1]), [interesting(0.5),file(topreal1,i2_24_4_1__topreal1),[file(topreal1,i2_24_4_1__topreal1)]]). fof(i1_24_4_1__topreal1,plain,( ~ ( r2_hidden(c1_24_4_1__topreal1,a_0_3_topreal1) & ~ r2_hidden(c1_24_4_1__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_24_4_1__topreal1]),discharge_asm(discharge,[e1_24_4_1__topreal1])],[e1_24_4_1__topreal1,i2_24_4_1__topreal1]), [interesting(0.5),file(topreal1,i1_24_4_1__topreal1),[file(topreal1,i1_24_4_1__topreal1)]]). fof(i1_24_4_1_tmp__topreal1,plain,( ~ ( r2_hidden(c1_24_4_1__topreal1,a_0_3_topreal1) & ~ r2_hidden(c1_24_4_1__topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_24_4_1__topreal1])],[dt_c1_24_4_1__topreal1,i1_24_4_1__topreal1]), [interesting(0.65),e1_24_4__topreal1]). fof(e1_24_4__topreal1,plain,( r1_tarski(a_0_3_topreal1,k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1))) ), inference(let,[status(thm),assumptions([])],[i1_24_4_1_tmp__topreal1,free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,abstractness_v1_pre_topc,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_struct_0,fc5_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,commutativity_k3_topreal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t2_tarski,fraenkel_a_0_3_topreal1,d3_tarski,dh_c1_24_4_1__topreal1]), [interesting(0.65),file(topreal1,e1_24_4__topreal1),[file(topreal1,e1_24_4__topreal1)]]). fof(e3_24_4__topreal1,plain,( a_0_3_topreal1 = k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)) ), inference(mizar_by,[status(thm),assumptions([])],[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_k6_xcmplx_0,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_relset_1,dt_u1_metric_1,cc1_relset_1,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc20_xreal_0,fc4_int_1,fc4_subset_1,fc5_xreal_0,fc8_int_1,fc9_int_1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,t4_arithm,free_g1_metric_1,free_g1_pre_topc,commutativity_k17_euclid,abstractness_v1_metric_1,existence_l1_metric_1,redefinition_k5_real_1,dt_g1_metric_1,dt_g1_pre_topc,dt_k13_euclid,dt_k17_euclid,dt_k18_euclid,dt_k1_euclid,dt_k5_real_1,dt_l1_metric_1,dt_u1_pre_topc,fc3_pcomps_1,fc4_pcomps_1,rc1_metric_1,rc2_metric_1,rc3_metric_1,d1_euclid,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,dt_k14_euclid,dt_k1_xboole_0,dt_k5_ordinal2,dt_k5_pcomps_1,dt_l1_pre_topc,dt_l1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_euclid,fc1_struct_0,fc1_xboole_0,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_pre_topc,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_pcomps_1,rc2_pre_topc,rc2_xreal_0,rc3_nat_1,rc3_struct_0,rc3_xreal_0,rc4_xreal_0,rc5_struct_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,fraenkel_a_3_0_topreal1,d7_euclid,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_finseq_1,dt_k15_euclid,dt_k1_numbers,dt_k1_topreal1,dt_k1_zfmisc_1,dt_k21_euclid,dt_k22_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_subset_1,fc1_topreal1,fc2_euclid,fc2_membered,fc2_topreal1,rc1_subset_1,rc1_xboole_0,rc2_subset_1,rc2_xboole_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,t1_numerals,t1_real,t1_subset,t2_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,d8_euclid,d3_topreal1,commutativity_k3_topreal1,reflexivity_r1_tarski,redefinition_k3_topreal1,dt_k23_euclid,dt_k3_topreal1,t3_subset,t2_tarski,fraenkel_a_0_3_topreal1,d16_euclid,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_24_4__topreal1,e1_24_4__topreal1,d10_xboole_0]), [interesting(0.65),file(topreal1,e3_24_4__topreal1),[file(topreal1,e3_24_4__topreal1)]]). fof(i1_24_4__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i1_24_4__topreal1)]), [interesting(0.65),trivial,file(topreal1,i1_24_4__topreal1)]). fof(e7_24__topreal1,plain,( a_0_3_topreal1 = k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)) ), inference(conclusion,[status(thm),assumptions([])],[e3_24_4__topreal1,i1_24_4__topreal1]), [interesting(0.8),file(topreal1,e7_24__topreal1),[file(topreal1,e7_24__topreal1)]]). fof(i4_24__topreal1,theorem,( $true ), introduced(tautology,[file(topreal1,i4_24__topreal1)]), [interesting(0.8),trivial,file(topreal1,i4_24__topreal1)]). fof(i3_24__topreal1,plain,( k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)) = a_0_3_topreal1 ), inference(conclusion,[status(thm),assumptions([])],[e7_24__topreal1,i4_24__topreal1]), [interesting(0.8),file(topreal1,i3_24__topreal1),[file(topreal1,i3_24__topreal1)]]). fof(i2_24__topreal1,plain, ( k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0)) = a_0_2_topreal1 & k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)) = a_0_3_topreal1 ), inference(conclusion,[status(thm),assumptions([])],[e6_24__topreal1,i3_24__topreal1]), [interesting(0.8),file(topreal1,i2_24__topreal1),[file(topreal1,i2_24__topreal1)]]). fof(i1_24__topreal1,plain, ( k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1)) = a_0_1_topreal1 & k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0)) = a_0_2_topreal1 & k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)) = a_0_3_topreal1 ), inference(conclusion,[status(thm),assumptions([])],[e5_24__topreal1,i2_24__topreal1]), [interesting(0.8),file(topreal1,i1_24__topreal1),[file(topreal1,i1_24__topreal1)]]). fof(t19_topreal1,theorem, ( k3_topreal1(2,k23_euclid(0,0),k23_euclid(0,1)) = a_0_0_topreal1 & k3_topreal1(2,k23_euclid(0,1),k23_euclid(1,1)) = a_0_1_topreal1 & k3_topreal1(2,k23_euclid(0,0),k23_euclid(1,0)) = a_0_2_topreal1 & k3_topreal1(2,k23_euclid(1,0),k23_euclid(1,1)) = a_0_3_topreal1 ), inference(conclusion,[status(thm),assumptions([])],[e4_24__topreal1,i1_24__topreal1]), [interesting(1),file(topreal1,t19_topreal1),[file(topreal1,t19_topreal1)]]).