% Mizar ND problem: t39_toprns_1,toprns_1,613,54 fof(dh_c1_45__toprns_1,definition, ( ( m2_subset_1(c1_45__toprns_1,k1_numbers,k5_numbers) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(D,B) & r1_xreal_0(C,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),A,D))) ) ) ) ) ) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ! [F] : ( ( v1_funct_1(F) & v1_funct_2(F,k5_numbers,u1_struct_0(k15_euclid(E))) & m2_relset_1(F,k5_numbers,u1_struct_0(k15_euclid(E))) ) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ? [H] : ( m1_subset_1(H,k1_numbers) & ~ r1_xreal_0(H,0) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(I,G) & r1_xreal_0(H,k5_toprns_1(E,k2_normsp_1(k15_euclid(E),F,I))) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_45__toprns_1),file(toprns_1,c1_45__toprns_1)]), [interesting(0.8),axiom,file(toprns_1,c1_45__toprns_1)]). fof(dh_c2_45__toprns_1,definition, ( ( ( v1_funct_1(c2_45__toprns_1) & v1_funct_2(c2_45__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(c2_45__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m1_subset_1(B,k1_numbers) & ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,A) & r1_xreal_0(B,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) ) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(D,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [F] : ( m1_subset_1(F,k1_numbers) & ~ r1_xreal_0(F,0) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(G,E) & r1_xreal_0(F,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),D,G))) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_45__toprns_1),file(toprns_1,c2_45__toprns_1)]), [interesting(0.8),axiom,file(toprns_1,c2_45__toprns_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(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_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_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(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(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_arytm_3,theorem,( ! [A] : ( v3_ordinal1(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) ) ) ) ), file(arytm_3,cc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc1_arytm_3)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). 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(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_arytm_3,theorem,( ! [A] : ( ( v1_xboole_0(A) & v3_ordinal1(A) ) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc2_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc2_arytm_3)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_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(rc1_arytm_3,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ), file(arytm_3,rc1_arytm_3), [interesting(0.9),axiom,file(arytm_3,rc1_arytm_3)]). fof(rc1_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_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_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_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(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(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_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). 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_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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(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_struct_0,axiom,( $true ), file(struct_0,u1_struct_0), [interesting(0.9),axiom,file(struct_0,u1_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(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(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(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_arytm_3,theorem,( ! [A] : ( m1_subset_1(A,k5_ordinal2) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) ) ) ), file(arytm_3,cc3_arytm_3), [interesting(0.9),axiom,file(arytm_3,cc3_arytm_3)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(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_ordinal2,theorem, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]). 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(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(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(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(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(redefinition_k2_normsp_1,definition,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => k2_normsp_1(A,B,C) = k1_funct_1(B,C) ) ), file(normsp_1,k2_normsp_1), [interesting(0.9),axiom,file(normsp_1,k2_normsp_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_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_k2_normsp_1,axiom,( ! [A,B,C] : ( ( ~ v3_struct_0(A) & l1_struct_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(A)) & m1_relset_1(B,k5_numbers,u1_struct_0(A)) & m1_subset_1(C,k5_numbers) ) => m1_subset_1(k2_normsp_1(A,B,C),u1_struct_0(A)) ) ), file(normsp_1,k2_normsp_1), [interesting(0.9),axiom,file(normsp_1,k2_normsp_1)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_k5_toprns_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,u1_struct_0(k15_euclid(A))) ) => m1_subset_1(k5_toprns_1(A,B),k1_numbers) ) ), file(toprns_1,k5_toprns_1), [interesting(0.9),axiom,file(toprns_1,k5_toprns_1)]). 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_c1_45__toprns_1,assumption,( m2_subset_1(c1_45__toprns_1,k1_numbers,k5_numbers) ), introduced(assumption,[file(toprns_1,c1_45__toprns_1)]), [interesting(0.8),axiom,file(toprns_1,c1_45__toprns_1)]). fof(dt_c2_45__toprns_1,assumption, ( v1_funct_1(c2_45__toprns_1) & v1_funct_2(c2_45__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(c2_45__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ), introduced(assumption,[file(toprns_1,c2_45__toprns_1)]), [interesting(0.8),axiom,file(toprns_1,c2_45__toprns_1)]). 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(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(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(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(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(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). 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(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(commutativity_k1_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(redefinition_k1_nat_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). 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(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(s1_nat_1__e3_45__toprns_1,theorem,( ! [A,B] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A))) & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) ) => ( ( ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(D,0) & r1_xreal_0(C,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,D))) ) ) ) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ? [F] : ( m1_subset_1(F,k1_numbers) & ~ r1_xreal_0(F,0) & ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(G,E) & r1_xreal_0(F,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,G))) ) ) ) => ? [H] : ( m1_subset_1(H,k1_numbers) & ~ r1_xreal_0(H,0) & ! [I] : ( m2_subset_1(I,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(I,k1_nat_1(E,1)) & r1_xreal_0(H,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,I))) ) ) ) ) ) ) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ? [J] : ( m1_subset_1(J,k1_numbers) & ~ r1_xreal_0(J,0) & ! [K] : ( m2_subset_1(K,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(K,E) & r1_xreal_0(J,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,K))) ) ) ) ) ) ) ), file(toprns_1,s1_nat_1__e3_45__toprns_1), [interesting(0.9),axiom,file(toprns_1,s1_nat_1__e3_45__toprns_1)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). 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(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(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(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(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(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(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(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(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(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__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(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). 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(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(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(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(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_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(de_c1_45_1__toprns_1,definition,( c1_45_1__toprns_1 = k3_real_1(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,0)),1) ), introduced(definition,[new_symbol(c1_45_1__toprns_1),file(toprns_1,c1_45_1__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,c1_45_1__toprns_1)]). fof(dt_c1_45_1__toprns_1,plain,( m1_subset_1(c1_45_1__toprns_1,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc3_xreal_0,fc5_membered,fc6_membered,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc5_struct_0,spc6_arithm,t1_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc4_membered,fc1_struct_0,fc2_euclid,rc3_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,existence_m1_subset_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k3_real_1,dt_k5_toprns_1,dt_m1_subset_1,dt_c1_45__toprns_1,dt_c2_45__toprns_1,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,de_c1_45_1__toprns_1]), [interesting(0.65),file(toprns_1,c1_45_1__toprns_1),[file(toprns_1,c1_45_1__toprns_1)]]). 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(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(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(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(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(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(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_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(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(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(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(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t4_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(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(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(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_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__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_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__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_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(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(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_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(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(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__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_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_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__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(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_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(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__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(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(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(t26_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m1_subset_1(B,u1_struct_0(k15_euclid(A))) => r1_xreal_0(0,k5_toprns_1(A,B)) ) ) ), file(toprns_1,t26_toprns_1), [interesting(0.9),axiom,file(toprns_1,t26_toprns_1)]). fof(e1_45_1__toprns_1,plain,( r1_xreal_0(0,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,0))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_45__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,spc0_numerals,spc0_boole,t26_toprns_1]), [interesting(0.65),file(toprns_1,e1_45_1__toprns_1),[file(toprns_1,e1_45_1__toprns_1)]]). fof(t10_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ~ ( ~ r1_xreal_0(B,A) & r1_xreal_0(C,D) & r1_xreal_0(k2_xcmplx_0(B,D),k2_xcmplx_0(A,C)) ) ) ) ) ) ), file(xreal_1,t10_xreal_1), [interesting(0.9),axiom,file(xreal_1,t10_xreal_1)]). 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(e2_45_1__toprns_1,plain,( ~ r1_xreal_0(k3_real_1(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,0)),1),k1_nat_1(0,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k5_toprns_1,dt_c1_45__toprns_1,dt_c2_45__toprns_1,cc2_xreal_0,fc3_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_45_1__toprns_1,t10_xreal_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(toprns_1,e2_45_1__toprns_1),[file(toprns_1,e2_45_1__toprns_1)]]). 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(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(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(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(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__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(e3_45_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_1__toprns_1,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_1__toprns_1,dt_c2_45__toprns_1,de_c1_45_1__toprns_1,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_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_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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,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_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_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,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_45_1__toprns_1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0]), [interesting(0.65),file(toprns_1,e3_45_1__toprns_1),[file(toprns_1,e3_45_1__toprns_1)]]). fof(dh_c2_45_1__toprns_1,definition, ( ( m2_subset_1(c2_45_1__toprns_1,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_45_1__toprns_1,0) & r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_1__toprns_1))) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,0) & r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), introduced(definition,[new_symbol(c2_45_1__toprns_1),file(toprns_1,c2_45_1__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,c2_45_1__toprns_1)]). fof(e4_45_1__toprns_1,assumption,( r1_xreal_0(c2_45_1__toprns_1,0) ), introduced(assumption,[file(toprns_1,e4_45_1__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,e4_45_1__toprns_1)]). fof(dt_c2_45_1__toprns_1,assumption,( m2_subset_1(c2_45_1__toprns_1,k1_numbers,k5_numbers) ), introduced(assumption,[file(toprns_1,c2_45_1__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,c2_45_1__toprns_1)]). fof(t18_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => r1_xreal_0(0,A) ) ), file(nat_1,t18_nat_1), [interesting(0.9),axiom,file(nat_1,t18_nat_1)]). fof(e5_45_1__toprns_1,plain,( 0 = c2_45_1__toprns_1 ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_1__toprns_1,e4_45_1__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c2_45_1__toprns_1,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_numerals,spc0_boole,e4_45_1__toprns_1,t18_nat_1]), [interesting(0.65),file(toprns_1,e5_45_1__toprns_1),[file(toprns_1,e5_45_1__toprns_1)]]). fof(e6_45_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_1__toprns_1,k3_real_1(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_1__toprns_1)),0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1,dt_c2_45_1__toprns_1,e4_45_1__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_1__toprns_1,dt_c2_45__toprns_1,dt_c2_45_1__toprns_1,de_c1_45_1__toprns_1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e5_45_1__toprns_1,t10_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(toprns_1,e6_45_1__toprns_1),[file(toprns_1,e6_45_1__toprns_1)]]). fof(e7_45_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_1__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1,dt_c2_45_1__toprns_1,e4_45_1__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_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,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k2_normsp_1,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_1__toprns_1,dt_c2_45__toprns_1,dt_c2_45_1__toprns_1,de_c1_45_1__toprns_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e6_45_1__toprns_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.65),file(toprns_1,e7_45_1__toprns_1),[file(toprns_1,e7_45_1__toprns_1)]]). fof(i5_45_1__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i5_45_1__toprns_1)]), [interesting(0.65),trivial,file(toprns_1,i5_45_1__toprns_1)]). fof(i4_45_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_1__toprns_1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1,dt_c2_45_1__toprns_1,e4_45_1__toprns_1])],[e7_45_1__toprns_1,i5_45_1__toprns_1]), [interesting(0.65),file(toprns_1,i4_45_1__toprns_1),[file(toprns_1,i4_45_1__toprns_1)]]). fof(i3_45_1__toprns_1,plain,( ~ ( r1_xreal_0(c2_45_1__toprns_1,0) & r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_1__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1,dt_c2_45_1__toprns_1]),discharge_asm(discharge,[e4_45_1__toprns_1])],[e4_45_1__toprns_1,i4_45_1__toprns_1]), [interesting(0.65),file(toprns_1,i3_45_1__toprns_1),[file(toprns_1,i3_45_1__toprns_1)]]). fof(i3_45_1_tmp__toprns_1,plain, ( m2_subset_1(c2_45_1__toprns_1,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_45_1__toprns_1,0) & r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_1__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1]),discharge_asm(discharge,[dt_c2_45_1__toprns_1])],[dt_c2_45_1__toprns_1,i3_45_1__toprns_1]), [interesting(0.65),i2_45_1__toprns_1]). fof(i2_45_1__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,0) & r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(let,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[i3_45_1_tmp__toprns_1,dh_c2_45_1__toprns_1]), [interesting(0.65),file(toprns_1,i2_45_1__toprns_1),[file(toprns_1,i2_45_1__toprns_1)]]). fof(i1_45_1__toprns_1,plain, ( ~ r1_xreal_0(c1_45_1__toprns_1,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,0) & r1_xreal_0(c1_45_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[e3_45_1__toprns_1,i2_45_1__toprns_1]), [interesting(0.65),file(toprns_1,i1_45_1__toprns_1),[file(toprns_1,i1_45_1__toprns_1)]]). fof(e1_45__toprns_1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,0) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(take,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_1__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc0_boole,i1_45_1__toprns_1]), [interesting(0.8),file(toprns_1,e1_45__toprns_1),[file(toprns_1,e1_45__toprns_1)]]). fof(dh_c1_45_2__toprns_1,definition, ( ( m2_subset_1(c1_45_2__toprns_1,k1_numbers,k5_numbers) => ( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,c1_45_2__toprns_1) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) => ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ? [D] : ( m1_subset_1(D,k1_numbers) & ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(E,C) & r1_xreal_0(D,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,E))) ) ) ) => ? [D] : ( m1_subset_1(D,k1_numbers) & ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(E,k1_nat_1(C,1)) & r1_xreal_0(D,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,E))) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_45_2__toprns_1),file(toprns_1,c1_45_2__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,c1_45_2__toprns_1)]). fof(dt_c1_45_2__toprns_1,assumption,( m2_subset_1(c1_45_2__toprns_1,k1_numbers,k5_numbers) ), introduced(assumption,[file(toprns_1,c1_45_2__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,c1_45_2__toprns_1)]). fof(e1_45_2__toprns_1,assumption,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,c1_45_2__toprns_1) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), introduced(assumption,[file(toprns_1,e1_45_2__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,e1_45_2__toprns_1)]). fof(dh_c2_45_2__toprns_1,definition, ( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,c1_45_2__toprns_1) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) => ( m1_subset_1(c2_45_2__toprns_1,k1_numbers) & ~ r1_xreal_0(c2_45_2__toprns_1,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,c1_45_2__toprns_1) & r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) ), introduced(definition,[new_symbol(c2_45_2__toprns_1),file(toprns_1,c2_45_2__toprns_1)]), [interesting(0.65),axiom,file(toprns_1,c2_45_2__toprns_1)]). fof(dt_c2_45_2__toprns_1,plain,( m1_subset_1(c2_45_2__toprns_1,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc0_boole,dh_c2_45_2__toprns_1,e1_45_2__toprns_1]), [interesting(0.65),file(toprns_1,c2_45_2__toprns_1),[file(toprns_1,c2_45_2__toprns_1)]]). fof(e1_45_2_2__toprns_1,assumption,( r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)))) ), introduced(assumption,[file(toprns_1,e1_45_2_2__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,e1_45_2_2__toprns_1)]). fof(de_c1_45_2_2__toprns_1,definition,( c1_45_2_2__toprns_1 = k3_real_1(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1))),1) ), introduced(definition,[new_symbol(c1_45_2_2__toprns_1),file(toprns_1,c1_45_2_2__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,c1_45_2_2__toprns_1)]). fof(dt_c1_45_2_2__toprns_1,plain,( m1_subset_1(c1_45_2_2__toprns_1,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc3_xreal_0,fc5_membered,fc6_membered,fc8_xreal_0,rc1_membered,rc1_xreal_0,rc5_struct_0,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc4_membered,fc1_struct_0,fc2_euclid,rc3_struct_0,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_real_1,existence_m1_subset_1,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k3_real_1,dt_k5_toprns_1,dt_m1_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,fc2_membered,spc1_numerals,spc1_boole,de_c1_45_2_2__toprns_1]), [interesting(0.5),file(toprns_1,c1_45_2_2__toprns_1),[file(toprns_1,c1_45_2_2__toprns_1)]]). fof(e2_45_2_2__toprns_1,plain,( r1_xreal_0(0,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_arithm,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,t1_numerals,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t26_toprns_1]), [interesting(0.5),file(toprns_1,e2_45_2_2__toprns_1),[file(toprns_1,e2_45_2_2__toprns_1)]]). fof(e3_45_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k1_nat_1(0,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k3_real_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k3_real_1,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,de_c1_45_2_2__toprns_1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,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_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_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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,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_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_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,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_45_2_2__toprns_1,t10_xreal_1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(toprns_1,e3_45_2_2__toprns_1),[file(toprns_1,e3_45_2_2__toprns_1)]]). fof(e4_45_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc1_arytm_3,cc1_relset_1,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc1_struct_0,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_0,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,commutativity_k3_real_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k3_real_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,cc15_membered,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,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_45_2_2__toprns_1,de_c1_45_2_2__toprns_1,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_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_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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,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_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_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,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_45_2_2__toprns_1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0]), [interesting(0.5),file(toprns_1,e4_45_2_2__toprns_1),[file(toprns_1,e4_45_2_2__toprns_1)]]). fof(dh_c2_45_2_2__toprns_1,definition, ( ( m2_subset_1(c2_45_2_2__toprns_1,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_45_2_2__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), introduced(definition,[new_symbol(c2_45_2_2__toprns_1),file(toprns_1,c2_45_2_2__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,c2_45_2_2__toprns_1)]). fof(e5_45_2_2__toprns_1,assumption,( r1_xreal_0(c2_45_2_2__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) ), introduced(assumption,[file(toprns_1,e5_45_2_2__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,e5_45_2_2__toprns_1)]). fof(dt_c2_45_2_2__toprns_1,assumption,( m2_subset_1(c2_45_2_2__toprns_1,k1_numbers,k5_numbers) ), introduced(assumption,[file(toprns_1,c2_45_2_2__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,c2_45_2_2__toprns_1)]). fof(e1_45_2_2_2__toprns_1,assumption,( c2_45_2_2__toprns_1 = k1_nat_1(c1_45_2__toprns_1,1) ), introduced(assumption,[file(toprns_1,e1_45_2_2_2__toprns_1)]), [interesting(0.35),axiom,file(toprns_1,e1_45_2_2_2__toprns_1)]). fof(e2_45_2_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k3_real_1(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1)),0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,e1_45_2_2_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,de_c1_45_2_2__toprns_1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_45_2_2_2__toprns_1,t10_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(toprns_1,e2_45_2_2_2__toprns_1),[file(toprns_1,e2_45_2_2_2__toprns_1)]]). fof(e3_45_2_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,e1_45_2_2_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_45_2__toprns_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_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,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k2_normsp_1,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,de_c1_45_2_2__toprns_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e2_45_2_2_2__toprns_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.35),file(toprns_1,e3_45_2_2_2__toprns_1),[file(toprns_1,e3_45_2_2_2__toprns_1)]]). fof(i2_45_2_2_2__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i2_45_2_2_2__toprns_1)]), [interesting(0.35),trivial,file(toprns_1,i2_45_2_2_2__toprns_1)]). fof(i1_45_2_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,e1_45_2_2_2__toprns_1])],[e3_45_2_2_2__toprns_1,i2_45_2_2_2__toprns_1]), [interesting(0.35),file(toprns_1,i1_45_2_2_2__toprns_1),[file(toprns_1,i1_45_2_2_2__toprns_1)]]). fof(e7_45_2_2__toprns_1,plain,( ~ ( c2_45_2_2__toprns_1 = k1_nat_1(c1_45_2__toprns_1,1) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1]),discharge_asm(discharge,[e1_45_2_2_2__toprns_1])],[e1_45_2_2_2__toprns_1,i1_45_2_2_2__toprns_1]), [interesting(0.5),file(toprns_1,e7_45_2_2__toprns_1),[file(toprns_1,e7_45_2_2__toprns_1)]]). fof(e1_45_2_2_1__toprns_1,assumption,( r1_xreal_0(c2_45_2_2__toprns_1,c1_45_2__toprns_1) ), introduced(assumption,[file(toprns_1,e1_45_2_2_1__toprns_1)]), [interesting(0.35),axiom,file(toprns_1,e1_45_2_2_1__toprns_1)]). fof(e3_45_2__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,c1_45_2__toprns_1) & r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc0_boole,dh_c2_45_2__toprns_1,e1_45_2__toprns_1]), [interesting(0.65),file(toprns_1,e3_45_2__toprns_1),[file(toprns_1,e3_45_2__toprns_1)]]). fof(e2_45_2_2_1__toprns_1,plain,( ~ r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,e1_45_2_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc2_euclid,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,dt_c2_45_2_2__toprns_1,fc2_membered,e1_45_2_2_1__toprns_1,e3_45_2__toprns_1]), [interesting(0.35),file(toprns_1,e2_45_2_2_1__toprns_1),[file(toprns_1,e2_45_2_2_1__toprns_1)]]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]). fof(e3_45_2_2_1__toprns_1,plain,( ~ r1_xreal_0(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1))),k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,e1_45_2_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k5_toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,dt_c2_45_2_2__toprns_1,cc2_xreal_0,spc1_numerals,spc1_boole,e2_45_2_2_1__toprns_1,e1_45_2_2__toprns_1,t2_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(toprns_1,e3_45_2_2_1__toprns_1),[file(toprns_1,e3_45_2_2_1__toprns_1)]]). fof(e4_45_2_2_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k3_real_1(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1)),0)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,e1_45_2_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,de_c1_45_2_2__toprns_1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e3_45_2_2_1__toprns_1,t10_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(toprns_1,e4_45_2_2_1__toprns_1),[file(toprns_1,e4_45_2_2_1__toprns_1)]]). fof(e5_45_2_2_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,e1_45_2_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_45_2__toprns_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_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,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k2_normsp_1,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,de_c1_45_2_2__toprns_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e4_45_2_2_1__toprns_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.35),file(toprns_1,e5_45_2_2_1__toprns_1),[file(toprns_1,e5_45_2_2_1__toprns_1)]]). fof(i2_45_2_2_1__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i2_45_2_2_1__toprns_1)]), [interesting(0.35),trivial,file(toprns_1,i2_45_2_2_1__toprns_1)]). fof(i1_45_2_2_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(conclusion,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,e1_45_2_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[e5_45_2_2_1__toprns_1,i2_45_2_2_1__toprns_1]), [interesting(0.35),file(toprns_1,i1_45_2_2_1__toprns_1),[file(toprns_1,i1_45_2_2_1__toprns_1)]]). fof(e6_45_2_2__toprns_1,plain,( ~ ( r1_xreal_0(c2_45_2_2__toprns_1,c1_45_2__toprns_1) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1]),discharge_asm(discharge,[e1_45_2_2_1__toprns_1])],[e1_45_2_2_1__toprns_1,i1_45_2_2_1__toprns_1]), [interesting(0.5),file(toprns_1,e6_45_2_2__toprns_1),[file(toprns_1,e6_45_2_2__toprns_1)]]). fof(t26_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,k2_xcmplx_0(B,1)) & ~ r1_xreal_0(A,B) & A != k2_xcmplx_0(B,1) ) ) ) ), file(nat_1,t26_nat_1), [interesting(0.9),axiom,file(nat_1,t26_nat_1)]). fof(e8_45_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([e5_45_2_2__toprns_1,dt_c2_45_2_2__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k3_real_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k3_real_1,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_2__toprns_1,de_c1_45_2_2__toprns_1,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e7_45_2_2__toprns_1,e5_45_2_2__toprns_1,e6_45_2_2__toprns_1,t26_nat_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(toprns_1,e8_45_2_2__toprns_1),[file(toprns_1,e8_45_2_2__toprns_1)]]). fof(i6_45_2_2__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i6_45_2_2__toprns_1)]), [interesting(0.5),trivial,file(toprns_1,i6_45_2_2__toprns_1)]). fof(i5_45_2_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ), inference(conclusion,[status(thm),assumptions([e5_45_2_2__toprns_1,dt_c2_45_2_2__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[e8_45_2_2__toprns_1,i6_45_2_2__toprns_1]), [interesting(0.5),file(toprns_1,i5_45_2_2__toprns_1),[file(toprns_1,i5_45_2_2__toprns_1)]]). fof(i4_45_2_2__toprns_1,plain,( ~ ( r1_xreal_0(c2_45_2_2__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_45_2_2__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1]),discharge_asm(discharge,[e5_45_2_2__toprns_1])],[e5_45_2_2__toprns_1,i5_45_2_2__toprns_1]), [interesting(0.5),file(toprns_1,i4_45_2_2__toprns_1),[file(toprns_1,i4_45_2_2__toprns_1)]]). fof(i4_45_2_2_tmp__toprns_1,plain, ( m2_subset_1(c2_45_2_2__toprns_1,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_45_2_2__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_2__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1]),discharge_asm(discharge,[dt_c2_45_2_2__toprns_1])],[dt_c2_45_2_2__toprns_1,i4_45_2_2__toprns_1]), [interesting(0.5),i3_45_2_2__toprns_1]). fof(i3_45_2_2__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(let,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[i4_45_2_2_tmp__toprns_1,dh_c2_45_2_2__toprns_1]), [interesting(0.5),file(toprns_1,i3_45_2_2__toprns_1),[file(toprns_1,i3_45_2_2__toprns_1)]]). fof(i2_45_2_2__toprns_1,plain, ( ~ r1_xreal_0(c1_45_2_2__toprns_1,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[e4_45_2_2__toprns_1,i3_45_2_2__toprns_1]), [interesting(0.5),file(toprns_1,i2_45_2_2__toprns_1),[file(toprns_1,i2_45_2_2__toprns_1)]]). fof(i1_45_2_2__toprns_1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(take,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_2__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_2__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,i2_45_2_2__toprns_1]), [interesting(0.5),file(toprns_1,i1_45_2_2__toprns_1),[file(toprns_1,i1_45_2_2__toprns_1)]]). fof(i1_45_2_2_tmp__toprns_1,plain, ( r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)))) => ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1]),discharge_asm(discharge,[e1_45_2_2__toprns_1])],[e1_45_2_2__toprns_1,i1_45_2_2__toprns_1]), [interesting(0.65),e5_45_2__toprns_1]). fof(e5_45_2__toprns_1,plain,( ~ ( r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)))) & ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[i1_45_2_2_tmp__toprns_1,free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole]), [interesting(0.65),file(toprns_1,e5_45_2__toprns_1),[file(toprns_1,e5_45_2__toprns_1)]]). fof(e1_45_2_1__toprns_1,assumption,( r1_xreal_0(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1))),c2_45_2__toprns_1) ), introduced(assumption,[file(toprns_1,e1_45_2_1__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,e1_45_2_1__toprns_1)]). fof(de_c1_45_2_1__toprns_1,definition,( c1_45_2_1__toprns_1 = k3_real_1(c2_45_2__toprns_1,1) ), introduced(definition,[new_symbol(c1_45_2_1__toprns_1),file(toprns_1,c1_45_2_1__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,c1_45_2_1__toprns_1)]). fof(dt_c1_45_2_1__toprns_1,plain,( m1_subset_1(c1_45_2_1__toprns_1,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc2_xreal_0,cc3_arytm_3,cc3_membered,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc3_xreal_0,fc5_membered,fc6_membered,fc8_xreal_0,rc1_membered,rc1_xreal_0,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc4_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,existence_m1_subset_1,redefinition_k3_real_1,dt_k1_numbers,dt_k3_real_1,dt_m1_subset_1,dt_c2_45_2__toprns_1,fc2_membered,spc1_numerals,spc1_boole,de_c1_45_2_1__toprns_1]), [interesting(0.5),file(toprns_1,c1_45_2_1__toprns_1),[file(toprns_1,c1_45_2_1__toprns_1)]]). fof(e2_45_2__toprns_1,plain,( ~ r1_xreal_0(c2_45_2__toprns_1,0) ), inference(consider,[status(thm),assumptions([e1_45_2__toprns_1])],[dh_c2_45_2__toprns_1,e1_45_2__toprns_1]), [interesting(0.65),file(toprns_1,e2_45_2__toprns_1),[file(toprns_1,e2_45_2__toprns_1)]]). fof(e2_45_2_1__toprns_1,plain,( ~ r1_xreal_0(k3_real_1(c2_45_2__toprns_1,1),k1_nat_1(0,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc2_membered,fc8_xreal_0,rc1_xreal_0,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k3_real_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_c2_45_2__toprns_1,cc2_xreal_0,fc3_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_45_2__toprns_1,t10_xreal_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(toprns_1,e2_45_2_1__toprns_1),[file(toprns_1,e2_45_2_1__toprns_1)]]). fof(e3_45_2_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,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_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k3_real_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_45_2_1__toprns_1,dt_c2_45_2__toprns_1,de_c1_45_2_1__toprns_1,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_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_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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,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_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_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,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e2_45_2_1__toprns_1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0]), [interesting(0.5),file(toprns_1,e3_45_2_1__toprns_1),[file(toprns_1,e3_45_2_1__toprns_1)]]). fof(dh_c2_45_2_1__toprns_1,definition, ( ( m2_subset_1(c2_45_2_1__toprns_1,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_45_2_1__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), introduced(definition,[new_symbol(c2_45_2_1__toprns_1),file(toprns_1,c2_45_2_1__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,c2_45_2_1__toprns_1)]). fof(e4_45_2_1__toprns_1,assumption,( r1_xreal_0(c2_45_2_1__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) ), introduced(assumption,[file(toprns_1,e4_45_2_1__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,e4_45_2_1__toprns_1)]). fof(dt_c2_45_2_1__toprns_1,assumption,( m2_subset_1(c2_45_2_1__toprns_1,k1_numbers,k5_numbers) ), introduced(assumption,[file(toprns_1,c2_45_2_1__toprns_1)]), [interesting(0.5),axiom,file(toprns_1,c2_45_2_1__toprns_1)]). fof(e1_45_2_1_2__toprns_1,assumption,( c2_45_2_1__toprns_1 = k1_nat_1(c1_45_2__toprns_1,1) ), introduced(assumption,[file(toprns_1,e1_45_2_1_2__toprns_1)]), [interesting(0.35),axiom,file(toprns_1,e1_45_2_1_2__toprns_1)]). fof(e2_45_2_1_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k3_real_1(c2_45_2__toprns_1,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_membered,fc8_xreal_0,rc1_xreal_0,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,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k3_real_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_45_2_1__toprns_1,dt_c2_45_2__toprns_1,de_c1_45_2_1__toprns_1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t10_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(toprns_1,e2_45_2_1_2__toprns_1),[file(toprns_1,e2_45_2_1_2__toprns_1)]]). fof(e3_45_2_1_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_1__toprns_1,e1_45_2_1_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_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,commutativity_k1_nat_1,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_1__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,dt_c2_45_2_1__toprns_1,de_c1_45_2_1__toprns_1,cc2_xreal_0,fc1_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e2_45_2_1_2__toprns_1,e1_45_2_1__toprns_1,e1_45_2_1_2__toprns_1,t2_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.35),file(toprns_1,e3_45_2_1_2__toprns_1),[file(toprns_1,e3_45_2_1_2__toprns_1)]]). fof(i2_45_2_1_2__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i2_45_2_1_2__toprns_1)]), [interesting(0.35),trivial,file(toprns_1,i2_45_2_1_2__toprns_1)]). fof(i1_45_2_1_2__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(conclusion,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_1__toprns_1,e1_45_2_1_2__toprns_1])],[e3_45_2_1_2__toprns_1,i2_45_2_1_2__toprns_1]), [interesting(0.35),file(toprns_1,i1_45_2_1_2__toprns_1),[file(toprns_1,i1_45_2_1_2__toprns_1)]]). fof(e6_45_2_1__toprns_1,plain,( ~ ( c2_45_2_1__toprns_1 = k1_nat_1(c1_45_2__toprns_1,1) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1,e1_45_2_1__toprns_1]),discharge_asm(discharge,[e1_45_2_1_2__toprns_1])],[e1_45_2_1_2__toprns_1,i1_45_2_1_2__toprns_1]), [interesting(0.5),file(toprns_1,e6_45_2_1__toprns_1),[file(toprns_1,e6_45_2_1__toprns_1)]]). fof(e1_45_2_1_1__toprns_1,assumption,( r1_xreal_0(c2_45_2_1__toprns_1,c1_45_2__toprns_1) ), introduced(assumption,[file(toprns_1,e1_45_2_1_1__toprns_1)]), [interesting(0.35),axiom,file(toprns_1,e1_45_2_1_1__toprns_1)]). fof(e3_45_2_1_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k3_real_1(c2_45_2__toprns_1,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc2_membered,fc8_xreal_0,rc1_xreal_0,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,commutativity_k2_xcmplx_0,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k3_real_1,dt_k2_xcmplx_0,dt_k3_real_1,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_45_2_1__toprns_1,dt_c2_45_2__toprns_1,de_c1_45_2_1__toprns_1,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t10_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(toprns_1,e3_45_2_1_1__toprns_1),[file(toprns_1,e3_45_2_1_1__toprns_1)]]). fof(e2_45_2_1_1__toprns_1,plain,( ~ r1_xreal_0(c2_45_2__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,e1_45_2_1_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc2_euclid,fc5_membered,rc3_struct_0,rc5_struct_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,dt_c2_45_2_1__toprns_1,fc2_membered,e1_45_2_1_1__toprns_1,e3_45_2__toprns_1]), [interesting(0.35),file(toprns_1,e2_45_2_1_1__toprns_1),[file(toprns_1,e2_45_2_1_1__toprns_1)]]). fof(e4_45_2_1_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,e1_45_2_1_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc1_xreal_0,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,cc15_membered,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_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,commutativity_k3_real_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k3_real_1,dt_k15_euclid,dt_k2_normsp_1,dt_k3_real_1,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2_1__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,dt_c2_45_2_1__toprns_1,de_c1_45_2_1__toprns_1,cc2_xreal_0,fc1_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e3_45_2_1_1__toprns_1,e2_45_2_1_1__toprns_1,t2_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.35),file(toprns_1,e4_45_2_1_1__toprns_1),[file(toprns_1,e4_45_2_1_1__toprns_1)]]). fof(i2_45_2_1_1__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i2_45_2_1_1__toprns_1)]), [interesting(0.35),trivial,file(toprns_1,i2_45_2_1_1__toprns_1)]). fof(i1_45_2_1_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(conclusion,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,e1_45_2_1_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[e4_45_2_1_1__toprns_1,i2_45_2_1_1__toprns_1]), [interesting(0.35),file(toprns_1,i1_45_2_1_1__toprns_1),[file(toprns_1,i1_45_2_1_1__toprns_1)]]). fof(e5_45_2_1__toprns_1,plain,( ~ ( r1_xreal_0(c2_45_2_1__toprns_1,c1_45_2__toprns_1) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1]),discharge_asm(discharge,[e1_45_2_1_1__toprns_1])],[e1_45_2_1_1__toprns_1,i1_45_2_1_1__toprns_1]), [interesting(0.5),file(toprns_1,e5_45_2_1__toprns_1),[file(toprns_1,e5_45_2_1__toprns_1)]]). fof(e7_45_2_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(mizar_by,[status(thm),assumptions([e1_45_2_1__toprns_1,e4_45_2_1__toprns_1,dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[reflexivity_r1_tarski,cc1_arytm_3,cc2_arytm_3,rc1_arytm_3,free_g1_pre_topc,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_u1_pre_topc,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_arytm_3,cc3_membered,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_ordinal2,fc20_xreal_0,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc2_xreal_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,commutativity_k3_real_1,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k3_real_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k3_real_1,dt_k5_numbers,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c2_45_2__toprns_1,cc15_membered,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_struct_0,fc1_xreal_0,fc2_euclid,fc2_membered,fc3_xreal_0,fc5_xreal_0,fc8_xreal_0,rc1_xreal_0,rc3_struct_0,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,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,dt_k15_euclid,dt_k1_nat_1,dt_k2_normsp_1,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k5_toprns_1,dt_k6_xcmplx_0,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_1__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2_1__toprns_1,de_c1_45_2_1__toprns_1,cc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e6_45_2_1__toprns_1,e4_45_2_1__toprns_1,e5_45_2_1__toprns_1,t26_nat_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(toprns_1,e7_45_2_1__toprns_1),[file(toprns_1,e7_45_2_1__toprns_1)]]). fof(i6_45_2_1__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i6_45_2_1__toprns_1)]), [interesting(0.5),trivial,file(toprns_1,i6_45_2_1__toprns_1)]). fof(i5_45_2_1__toprns_1,plain,( ~ r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ), inference(conclusion,[status(thm),assumptions([e1_45_2_1__toprns_1,e4_45_2_1__toprns_1,dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[e7_45_2_1__toprns_1,i6_45_2_1__toprns_1]), [interesting(0.5),file(toprns_1,i5_45_2_1__toprns_1),[file(toprns_1,i5_45_2_1__toprns_1)]]). fof(i4_45_2_1__toprns_1,plain,( ~ ( r1_xreal_0(c2_45_2_1__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([e1_45_2_1__toprns_1,dt_c2_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1]),discharge_asm(discharge,[e4_45_2_1__toprns_1])],[e4_45_2_1__toprns_1,i5_45_2_1__toprns_1]), [interesting(0.5),file(toprns_1,i4_45_2_1__toprns_1),[file(toprns_1,i4_45_2_1__toprns_1)]]). fof(i4_45_2_1_tmp__toprns_1,plain, ( m2_subset_1(c2_45_2_1__toprns_1,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(c2_45_2_1__toprns_1,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,c2_45_2_1__toprns_1))) ) ), inference(discharge_asm,[status(thm),assumptions([e1_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1]),discharge_asm(discharge,[dt_c2_45_2_1__toprns_1])],[dt_c2_45_2_1__toprns_1,i4_45_2_1__toprns_1]), [interesting(0.5),i3_45_2_1__toprns_1]). fof(i3_45_2_1__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(let,[status(thm),assumptions([e1_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[i4_45_2_1_tmp__toprns_1,dh_c2_45_2_1__toprns_1]), [interesting(0.5),file(toprns_1,i3_45_2_1__toprns_1),[file(toprns_1,i3_45_2_1__toprns_1)]]). fof(i2_45_2_1__toprns_1,plain, ( ~ r1_xreal_0(c1_45_2_1__toprns_1,0) & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(A,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(c1_45_2_1__toprns_1,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,A))) ) ) ), inference(conclusion,[status(thm),assumptions([e1_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[e3_45_2_1__toprns_1,i3_45_2_1__toprns_1]), [interesting(0.5),file(toprns_1,i2_45_2_1__toprns_1),[file(toprns_1,i2_45_2_1__toprns_1)]]). fof(i1_45_2_1__toprns_1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(take,[status(thm),assumptions([e1_45_2_1__toprns_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c1_45_2_1__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,i2_45_2_1__toprns_1]), [interesting(0.5),file(toprns_1,i1_45_2_1__toprns_1),[file(toprns_1,i1_45_2_1__toprns_1)]]). fof(i1_45_2_1_tmp__toprns_1,plain, ( r1_xreal_0(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1))),c2_45_2__toprns_1) => ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1]),discharge_asm(discharge,[e1_45_2_1__toprns_1])],[e1_45_2_1__toprns_1,i1_45_2_1__toprns_1]), [interesting(0.65),e4_45_2__toprns_1]). fof(e4_45_2__toprns_1,plain,( ~ ( r1_xreal_0(k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,k1_nat_1(c1_45_2__toprns_1,1))),c2_45_2__toprns_1) & ! [A] : ( m1_subset_1(A,k1_numbers) => ~ ( ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[i1_45_2_1_tmp__toprns_1,free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole]), [interesting(0.65),file(toprns_1,e4_45_2__toprns_1),[file(toprns_1,e4_45_2__toprns_1)]]). fof(e6_45_2__toprns_1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_g1_pre_topc,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_arithm,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,existence_l1_pre_topc,existence_l1_struct_0,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_real,t2_subset,t3_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,dt_c2_45_2__toprns_1,fc2_euclid,fc2_membered,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_numerals,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e5_45_2__toprns_1,e4_45_2__toprns_1,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.65),file(toprns_1,e6_45_2__toprns_1),[file(toprns_1,e6_45_2__toprns_1)]]). fof(i3_45_2__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i3_45_2__toprns_1)]), [interesting(0.65),trivial,file(toprns_1,i3_45_2__toprns_1)]). fof(i2_45_2__toprns_1,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,e1_45_2__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e6_45_2__toprns_1,i3_45_2__toprns_1]), [interesting(0.65),file(toprns_1,i2_45_2__toprns_1),[file(toprns_1,i2_45_2__toprns_1)]]). fof(i2_45_2_tmp__toprns_1,plain, ( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,c1_45_2__toprns_1) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) => ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(D,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(C,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,D))) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1]),discharge_asm(discharge,[e1_45_2__toprns_1])],[e1_45_2__toprns_1,i2_45_2__toprns_1]), [interesting(0.65),i1_45_2__toprns_1]). fof(i1_45_2__toprns_1,plain, ( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,c1_45_2__toprns_1) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) => ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1])],[i2_45_2_tmp__toprns_1,free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c1_45_2__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole]), [interesting(0.65),file(toprns_1,i1_45_2__toprns_1),[file(toprns_1,i1_45_2__toprns_1)]]). fof(i1_45_2_tmp__toprns_1,plain, ( m2_subset_1(c1_45_2__toprns_1,k1_numbers,k5_numbers) => ( ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,c1_45_2__toprns_1) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) => ? [A] : ( m1_subset_1(A,k1_numbers) & ~ r1_xreal_0(A,0) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(B,k1_nat_1(c1_45_2__toprns_1,1)) & r1_xreal_0(A,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,B))) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1]),discharge_asm(discharge,[dt_c1_45_2__toprns_1])],[dt_c1_45_2__toprns_1,i1_45_2__toprns_1]), [interesting(0.8),e2_45__toprns_1]). fof(e2_45__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ? [B] : ( m1_subset_1(B,k1_numbers) & ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,A) & r1_xreal_0(B,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) => ? [B] : ( m1_subset_1(B,k1_numbers) & ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,k1_nat_1(A,1)) & r1_xreal_0(B,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[i1_45_2_tmp__toprns_1,dh_c1_45_2__toprns_1]), [interesting(0.8),file(toprns_1,e2_45__toprns_1),[file(toprns_1,e2_45__toprns_1)]]). fof(e3_45__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m1_subset_1(B,k1_numbers) & ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,A) & r1_xreal_0(B,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc7_xreal_0,fc9_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,abstractness_v1_pre_topc,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc3_struct_0,rc5_struct_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_nat_1,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_u1_struct_0,dt_c1_45__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,s1_nat_1__e3_45__toprns_1,e1_45__toprns_1,e2_45__toprns_1]), [interesting(0.8),file(toprns_1,e3_45__toprns_1),[file(toprns_1,e3_45__toprns_1)]]). fof(i3_45__toprns_1,theorem,( $true ), introduced(tautology,[file(toprns_1,i3_45__toprns_1)]), [interesting(0.8),trivial,file(toprns_1,i3_45__toprns_1)]). fof(i2_45__toprns_1,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m1_subset_1(B,k1_numbers) & ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,A) & r1_xreal_0(B,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_45__toprns_1,dt_c2_45__toprns_1])],[free_g1_pre_topc,dt_g1_pre_topc,dt_k2_zfmisc_1,dt_u1_pre_topc,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_arytm_3,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_arytm_3,cc2_membered,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,rc1_arytm_3,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,abstractness_v1_pre_topc,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_l1_pre_topc,dt_l1_struct_0,dt_m1_relset_1,dt_m2_relset_1,dt_u1_struct_0,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc2_xreal_0,cc3_arytm_3,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc1_struct_0,fc5_membered,rc3_struct_0,rc5_struct_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_normsp_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k15_euclid,dt_k1_numbers,dt_k2_normsp_1,dt_k5_numbers,dt_k5_toprns_1,dt_m1_subset_1,dt_m2_subset_1,dt_c1_45__toprns_1,dt_c2_45__toprns_1,fc2_euclid,fc2_membered,spc0_numerals,spc0_boole,e3_45__toprns_1,i3_45__toprns_1]), [interesting(0.8),file(toprns_1,i2_45__toprns_1),[file(toprns_1,i2_45__toprns_1)]]). fof(i2_45_tmp__toprns_1,plain, ( ( v1_funct_1(c2_45__toprns_1) & v1_funct_2(c2_45__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(c2_45__toprns_1,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ? [B] : ( m1_subset_1(B,k1_numbers) & ~ r1_xreal_0(B,0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(C,A) & r1_xreal_0(B,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),c2_45__toprns_1,C))) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_45__toprns_1]),discharge_asm(discharge,[dt_c2_45__toprns_1])],[dt_c2_45__toprns_1,i2_45__toprns_1]), [interesting(0.8),i1_45__toprns_1]). fof(i1_45__toprns_1,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(D,B) & r1_xreal_0(C,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),A,D))) ) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_45__toprns_1])],[i2_45_tmp__toprns_1,dh_c2_45__toprns_1]), [interesting(0.8),file(toprns_1,i1_45__toprns_1),[file(toprns_1,i1_45__toprns_1)]]). fof(i1_45_tmp__toprns_1,plain, ( m2_subset_1(c1_45__toprns_1,k1_numbers,k5_numbers) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) & m2_relset_1(A,k5_numbers,u1_struct_0(k15_euclid(c1_45__toprns_1))) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ? [C] : ( m1_subset_1(C,k1_numbers) & ~ r1_xreal_0(C,0) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(D,B) & r1_xreal_0(C,k5_toprns_1(c1_45__toprns_1,k2_normsp_1(k15_euclid(c1_45__toprns_1),A,D))) ) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_45__toprns_1])],[dt_c1_45__toprns_1,i1_45__toprns_1]), [interesting(1),t39_toprns_1]). fof(t39_toprns_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,u1_struct_0(k15_euclid(A))) & m2_relset_1(B,k5_numbers,u1_struct_0(k15_euclid(A))) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ? [D] : ( m1_subset_1(D,k1_numbers) & ~ r1_xreal_0(D,0) & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ~ ( r1_xreal_0(E,C) & r1_xreal_0(D,k5_toprns_1(A,k2_normsp_1(k15_euclid(A),B,E))) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_45_tmp__toprns_1,dh_c1_45__toprns_1]), [interesting(1),file(toprns_1,t39_toprns_1),[file(toprns_1,t39_toprns_1)]]).