% Mizar ND problem: t3_binari_4,binari_4,71,38 fof(dh_c1_3__binari_4,definition, ( ( m2_subset_1(c1_3__binari_4,k1_numbers,k5_numbers) => k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)) = k5_euclid(c1_3__binari_4) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k7_euclid(A,k5_euclid(A),k5_euclid(A)) = k5_euclid(A) ) ), introduced(definition,[new_symbol(c1_3__binari_4),file(binari_4,c1_3__binari_4)]), [interesting(0.8),axiom,file(binari_4,c1_3__binari_4)]). 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(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(cc1_relset_1,theorem,( ! [A,B,C] : ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) => v1_relat_1(C) ) ), file(relset_1,cc1_relset_1), [interesting(0.9),axiom,file(relset_1,cc1_relset_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_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(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(rc1_margrel1,theorem,( ? [A] : v1_margrel1(A) ), file(margrel1,rc1_margrel1), [interesting(0.9),axiom,file(margrel1,rc1_margrel1)]). 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(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(redefinition_m2_finseq_1,definition,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(dt_m1_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(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_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(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_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), [interesting(0.9),axiom,file(int_1,cc4_int_1)]). fof(cc4_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(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(fc1_margrel1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_margrel1(k1_xboole_0) ), file(margrel1,fc1_margrel1), [interesting(0.9),axiom,file(margrel1,fc1_margrel1)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_int_1,theorem,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), [interesting(0.9),axiom,file(int_1,rc1_int_1)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_nat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_int_1,theorem,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), [interesting(0.9),axiom,file(int_1,rc2_int_1)]). fof(rc2_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_1)]). fof(rc3_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), [interesting(0.9),axiom,file(nat_1,rc3_nat_1)]). fof(commutativity_k3_rvsum_1,theorem,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_finseq_1(B,k1_numbers) ) => k3_rvsum_1(A,B) = k3_rvsum_1(B,A) ) ), file(rvsum_1,k3_rvsum_1), [interesting(0.9),axiom,file(rvsum_1,k3_rvsum_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(existence_m2_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ? [C] : m2_finseq_2(C,A,B) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(redefinition_m2_finseq_2,definition,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(dt_k1_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( ~ v1_xboole_0(k1_euclid(A)) & m1_finseq_2(k1_euclid(A),k1_numbers) ) ) ), file(euclid,k1_euclid), [interesting(0.9),axiom,file(euclid,k1_euclid)]). fof(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), [interesting(0.9),axiom,file(finseq_1,k1_finseq_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_k3_rvsum_1,axiom,( ! [A,B] : ( ( m1_finseq_1(A,k1_numbers) & m1_finseq_1(B,k1_numbers) ) => m2_finseq_1(k3_rvsum_1(A,B),k1_numbers) ) ), file(rvsum_1,k3_rvsum_1), [interesting(0.9),axiom,file(rvsum_1,k3_rvsum_1)]). fof(dt_k4_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_1(k4_euclid(A),k1_numbers) ) ), file(euclid,k4_euclid), [interesting(0.9),axiom,file(euclid,k4_euclid)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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_finseq_2,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(B) & m1_finseq_2(B,A) ) => ! [C] : ( m2_finseq_2(C,A,B) => m2_finseq_1(C,A) ) ) ), file(finseq_2,m2_finseq_2), [interesting(0.9),axiom,file(finseq_2,m2_finseq_2)]). fof(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(cc1_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), [interesting(0.9),axiom,file(nat_1,cc1_nat_1)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(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(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(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_k7_euclid,theorem,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k1_euclid(A)) ) => k7_euclid(A,B,C) = k7_euclid(A,C,B) ) ), file(euclid,k7_euclid), [interesting(0.9),axiom,file(euclid,k7_euclid)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k2_finseq_1,definition,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]). fof(redefinition_k4_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]). fof(redefinition_k5_euclid,definition,( ! [A] : ( m1_subset_1(A,k5_numbers) => k5_euclid(A) = k4_euclid(A) ) ), file(euclid,k5_euclid), [interesting(0.9),axiom,file(euclid,k5_euclid)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_k7_euclid,definition,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k1_euclid(A)) ) => k7_euclid(A,B,C) = k3_rvsum_1(B,C) ) ), file(euclid,k7_euclid), [interesting(0.9),axiom,file(euclid,k7_euclid)]). 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_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), [interesting(0.9),axiom,file(relat_1,k1_relat_1)]). fof(dt_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_1), [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]). fof(dt_k5_euclid,axiom,( ! [A] : ( m1_subset_1(A,k5_numbers) => m2_finseq_2(k5_euclid(A),k1_numbers,k1_euclid(A)) ) ), file(euclid,k5_euclid), [interesting(0.9),axiom,file(euclid,k5_euclid)]). fof(dt_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_k7_euclid,axiom,( ! [A,B,C] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k1_euclid(A)) & m1_subset_1(C,k1_euclid(A)) ) => m2_finseq_2(k7_euclid(A,B,C),k1_numbers,k1_euclid(A)) ) ), file(euclid,k7_euclid), [interesting(0.9),axiom,file(euclid,k7_euclid)]). 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_3__binari_4,assumption,( m2_subset_1(c1_3__binari_4,k1_numbers,k5_numbers) ), introduced(assumption,[file(binari_4,c1_3__binari_4)]), [interesting(0.8),axiom,file(binari_4,c1_3__binari_4)]). 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(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(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(dh_c1_3_3__binari_4,definition, ( ( m2_subset_1(c1_3_3__binari_4,k1_numbers,k5_numbers) => ( r2_hidden(c1_3_3__binari_4,k1_relat_1(k5_euclid(c1_3__binari_4))) => k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),c1_3_3__binari_4) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k1_relat_1(k5_euclid(c1_3__binari_4))) => k1_funct_1(k5_euclid(c1_3__binari_4),A) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),A) ) ) ), introduced(definition,[new_symbol(c1_3_3__binari_4),file(binari_4,c1_3_3__binari_4)]), [interesting(0.65),axiom,file(binari_4,c1_3_3__binari_4)]). fof(e1_3_3__binari_4,assumption,( r2_hidden(c1_3_3__binari_4,k1_relat_1(k5_euclid(c1_3__binari_4))) ), introduced(assumption,[file(binari_4,e1_3_3__binari_4)]), [interesting(0.65),axiom,file(binari_4,e1_3_3__binari_4)]). fof(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc1_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), [interesting(0.9),axiom,file(int_1,fc1_int_1)]). fof(fc1_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), [interesting(0.9),axiom,file(nat_1,fc1_nat_1)]). fof(fc3_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), [interesting(0.9),axiom,file(nat_1,fc3_nat_1)]). fof(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(fc4_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), [interesting(0.9),axiom,file(nat_1,fc4_nat_1)]). fof(fc6_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), [interesting(0.9),axiom,file(int_1,fc6_int_1)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(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(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(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_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). 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(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(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(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_c1_3_3__binari_4,assumption,( m2_subset_1(c1_3_3__binari_4,k1_numbers,k5_numbers) ), introduced(assumption,[file(binari_4,c1_3_3__binari_4)]), [interesting(0.65),axiom,file(binari_4,c1_3_3__binari_4)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r2_r2,theorem,( k2_xcmplx_0(0,2) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__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(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(commutativity_k9_binop_2,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k9_binop_2(B,A) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(redefinition_k2_seq_1,definition,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(redefinition_k9_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(dt_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k9_binop_2(A,B),k1_numbers) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(dt_k2_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => ( v1_relat_1(k2_finseq_2(A,B)) & v1_funct_1(k2_finseq_2(A,B)) & v1_finseq_1(k2_finseq_2(A,B)) ) ) ), file(finseq_2,k2_finseq_2), [interesting(0.9),axiom,file(finseq_2,k2_finseq_2)]). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), [interesting(0.9),axiom,file(finseq_2,k4_finseq_2)]). fof(redefinition_k4_finseqop,definition,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => k4_finseqop(A,B,C) = k2_finseq_2(B,C) ) ), file(finseqop,k4_finseqop), [interesting(0.9),axiom,file(finseqop,k4_finseqop)]). fof(dt_k4_finseqop,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,k5_numbers) & m1_subset_1(C,A) ) => m2_finseq_2(k4_finseqop(A,B,C),A,k4_finseq_2(B,A)) ) ), file(finseqop,k4_finseqop), [interesting(0.9),axiom,file(finseqop,k4_finseqop)]). fof(d4_euclid,definition,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k4_euclid(A) = k4_finseqop(k1_numbers,A,0) ) ), file(euclid,d4_euclid), [interesting(0.9),axiom,file(euclid,d4_euclid)]). fof(e1_3_3_1__binari_4,plain,( k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = k1_funct_1(k4_finseqop(k1_numbers,c1_3__binari_4,0),c1_3_3__binari_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4,dt_c1_3_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,rc1_margrel1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_finseq_2,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_margrel1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,redefinition_m2_finseq_1,redefinition_m2_finseq_2,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_finseq_2,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k4_finseqop,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k4_euclid,dt_k4_finseqop,dt_k5_euclid,dt_k5_numbers,dt_m2_subset_1,dt_c1_3__binari_4,dt_c1_3_3__binari_4,fc2_membered,spc0_boole,spc0_numerals,d4_euclid]), [interesting(0.5),file(binari_4,e1_3_3_1__binari_4),[file(binari_4,e1_3_3_1__binari_4)]]). fof(e1_3_2__binari_4,plain,( k1_relat_1(k5_euclid(c1_3__binari_4)) = k1_relat_1(k4_finseqop(k1_numbers,c1_3__binari_4,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,rc1_margrel1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_finseq_2,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_margrel1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,redefinition_m2_finseq_1,redefinition_m2_finseq_2,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_finseq_2,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k4_finseqop,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_relat_1,dt_k4_euclid,dt_k4_finseqop,dt_k5_euclid,dt_k5_numbers,dt_m2_subset_1,dt_c1_3__binari_4,fc2_membered,spc0_boole,spc0_numerals,d4_euclid]), [interesting(0.65),file(binari_4,e1_3_2__binari_4),[file(binari_4,e1_3_2__binari_4)]]). fof(t68_finseq_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : k4_finseq_1(k2_finseq_2(A,B)) = k2_finseq_1(A) ) ), file(finseq_2,t68_finseq_2), [interesting(0.9),axiom,file(finseq_2,t68_finseq_2)]). fof(e2_3_2__binari_4,plain,( k1_relat_1(k4_finseqop(k1_numbers,c1_3__binari_4,0)) = k2_finseq_1(c1_3__binari_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,rc1_margrel1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k4_finseq_2,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,redefinition_k2_finseq_1,redefinition_k4_finseq_1,redefinition_k4_finseqop,dt_k1_numbers,dt_k1_relat_1,dt_k2_finseq_1,dt_k2_finseq_2,dt_k4_finseq_1,dt_k4_finseqop,dt_c1_3__binari_4,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc2_membered,spc0_boole,spc0_numerals,t68_finseq_2]), [interesting(0.65),file(binari_4,e2_3_2__binari_4),[file(binari_4,e2_3_2__binari_4)]]). fof(e5_3__binari_4,plain,( k1_relat_1(k5_euclid(c1_3__binari_4)) = k2_finseq_1(c1_3__binari_4) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_3__binari_4])],[e1_3_2__binari_4,e2_3_2__binari_4]), [interesting(0.8),file(binari_4,e5_3__binari_4),[file(binari_4,e5_3__binari_4)]]). fof(t70_finseq_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B,C] : ( r2_hidden(B,k2_finseq_1(A)) => k1_funct_1(k2_finseq_2(A,C),B) = C ) ) ), file(finseq_2,t70_finseq_2), [interesting(0.9),axiom,file(finseq_2,t70_finseq_2)]). fof(e2_3_3_1__binari_4,plain,( k1_funct_1(k4_finseqop(k1_numbers,c1_3__binari_4,0),c1_3_3__binari_4) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_3_3__binari_4,dt_c1_3__binari_4,e1_3_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,rc1_margrel1,reflexivity_r1_tarski,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc2_nat_1,rc3_nat_1,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_euclid,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k4_euclid,dt_k4_finseq_2,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k2_finseq_1,redefinition_k4_finseqop,redefinition_k5_euclid,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_finseq_1,dt_k2_finseq_2,dt_k4_finseqop,dt_k5_euclid,dt_c1_3__binari_4,dt_c1_3_3__binari_4,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc2_membered,t1_subset,t7_boole,spc0_boole,spc0_numerals,e5_3__binari_4,e1_3_3__binari_4,t70_finseq_2]), [interesting(0.5),file(binari_4,e2_3_3_1__binari_4),[file(binari_4,e2_3_3_1__binari_4)]]). fof(e2_3_3__binari_4,plain,( k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = 0 ), inference(iterative_eq,[status(thm),assumptions([dt_c1_3_3__binari_4,dt_c1_3__binari_4,e1_3_3__binari_4])],[e1_3_3_1__binari_4,e2_3_3_1__binari_4]), [interesting(0.65),file(binari_4,e2_3_3__binari_4),[file(binari_4,e2_3_3__binari_4)]]). fof(dt_k3_funcop_1,axiom,( $true ), file(funcop_1,k3_funcop_1), [interesting(0.9),axiom,file(funcop_1,k3_funcop_1)]). fof(redefinition_k1_finseqop,definition,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_finseq_1(E,A) & m1_finseq_1(F,B) ) => k1_finseqop(A,B,C,D,E,F) = k3_funcop_1(D,E,F) ) ), file(finseqop,k1_finseqop), [interesting(0.9),axiom,file(finseqop,k1_finseqop)]). fof(dt_k1_finseqop,axiom,( ! [A,B,C,D,E,F] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & ~ v1_xboole_0(C) & v1_funct_1(D) & v1_funct_2(D,k2_zfmisc_1(A,B),C) & m1_relset_1(D,k2_zfmisc_1(A,B),C) & m1_finseq_1(E,A) & m1_finseq_1(F,B) ) => m2_finseq_1(k1_finseqop(A,B,C,D,E,F),C) ) ), file(finseqop,k1_finseqop), [interesting(0.9),axiom,file(finseqop,k1_finseqop)]). fof(dt_k33_binop_2,axiom, ( v1_funct_1(k33_binop_2) & v1_funct_2(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) & m2_relset_1(k33_binop_2,k2_zfmisc_1(k1_numbers,k1_numbers),k1_numbers) ), file(binop_2,k33_binop_2), [interesting(0.9),axiom,file(binop_2,k33_binop_2)]). fof(d4_rvsum_1,definition,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => k3_rvsum_1(A,B) = k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,A,B) ) ) ), file(rvsum_1,d4_rvsum_1), [interesting(0.9),axiom,file(rvsum_1,d4_rvsum_1)]). fof(e1_3_1__binari_4,plain,( k1_relat_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4))) = k1_relat_1(k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[reflexivity_r1_tarski,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_margrel1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_m1_finseq_2,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_euclid,dt_k2_zfmisc_1,dt_k3_funcop_1,dt_k4_euclid,dt_k5_numbers,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc4_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_rvsum_1,commutativity_k7_euclid,existence_m2_finseq_1,redefinition_k1_finseqop,redefinition_k5_euclid,redefinition_k7_euclid,redefinition_m2_finseq_1,dt_k1_finseqop,dt_k1_numbers,dt_k1_relat_1,dt_k33_binop_2,dt_k3_rvsum_1,dt_k5_euclid,dt_k7_euclid,dt_m2_finseq_1,dt_c1_3__binari_4,fc2_membered,d4_rvsum_1]), [interesting(0.65),file(binari_4,e1_3_1__binari_4),[file(binari_4,e1_3_1__binari_4)]]). fof(fc31_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), [interesting(0.9),axiom,file(membered,fc31_membered)]). fof(fc32_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), [interesting(0.9),axiom,file(membered,fc32_membered)]). fof(fc33_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), [interesting(0.9),axiom,file(membered,fc33_membered)]). fof(fc34_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), [interesting(0.9),axiom,file(membered,fc34_membered)]). fof(fc35_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), [interesting(0.9),axiom,file(membered,fc35_membered)]). fof(fc36_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), [interesting(0.9),axiom,file(membered,fc36_membered)]). fof(t2_boole,theorem,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), [interesting(0.9),axiom,file(boole,t2_boole)]). fof(fc27_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), [interesting(0.9),axiom,file(membered,fc27_membered)]). fof(fc28_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), [interesting(0.9),axiom,file(membered,fc28_membered)]). fof(fc29_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), [interesting(0.9),axiom,file(membered,fc29_membered)]). fof(fc30_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), [interesting(0.9),axiom,file(membered,fc30_membered)]). fof(commutativity_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(idempotence_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), [interesting(0.9),axiom,file(relat_1,k2_relat_1)]). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(d4_finseq_1,definition,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( m1_finseq_1(B,A) <=> r1_tarski(k2_relat_1(B),A) ) ) ), file(finseq_1,d4_finseq_1), [interesting(0.9),axiom,file(finseq_1,d4_finseq_1)]). fof(e2_3__binari_4,plain,( r1_tarski(k2_relat_1(k5_euclid(c1_3__binari_4)),k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m2_relset_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_margrel1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_margrel1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_euclid,dt_k1_zfmisc_1,dt_k4_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,existence_m1_finseq_1,redefinition_k5_euclid,dt_k1_numbers,dt_k2_relat_1,dt_k5_euclid,dt_m1_finseq_1,dt_c1_3__binari_4,fc2_membered,t3_subset,d4_finseq_1]), [interesting(0.8),file(binari_4,e2_3__binari_4),[file(binari_4,e2_3__binari_4)]]). fof(redefinition_k4_relset_1,definition,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(d1_funct_2,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) => ( ( ( B = k1_xboole_0 => A = k1_xboole_0 ) => ( v1_funct_2(C,A,B) <=> A = k4_relset_1(A,B,C) ) ) & ( B = k1_xboole_0 => ( A = k1_xboole_0 | ( v1_funct_2(C,A,B) <=> C = k1_xboole_0 ) ) ) ) ) ), file(funct_2,d1_funct_2), [interesting(0.9),axiom,file(funct_2,d1_funct_2)]). fof(e1_3__binari_4,plain,( k1_relat_1(k33_binop_2) = k2_zfmisc_1(k1_numbers,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc7_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc6_membered,rc1_margrel1,rc1_membered,t2_subset,t3_subset,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k4_relset_1,dt_m2_relset_1,fc1_margrel1,fc2_membered,fc6_membered,t6_boole,d1_funct_2]), [interesting(0.8),file(binari_4,e1_3__binari_4),[file(binari_4,e1_3__binari_4)]]). fof(t119_zfmisc_1,theorem,( ! [A,B,C,D] : ( ( r1_tarski(A,B) & r1_tarski(C,D) ) => r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ), file(zfmisc_1,t119_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,t119_zfmisc_1)]). fof(e3_3__binari_4,plain,( r1_tarski(k2_zfmisc_1(k2_relat_1(k5_euclid(c1_3__binari_4)),k2_relat_1(k5_euclid(c1_3__binari_4))),k1_relat_1(k33_binop_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_finseq_1,dt_m1_finseq_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_margrel1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m1_relset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_margrel1,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_euclid,dt_k1_zfmisc_1,dt_k4_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,t2_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_tarski,redefinition_k5_euclid,dt_k1_numbers,dt_k1_relat_1,dt_k2_relat_1,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k5_euclid,dt_c1_3__binari_4,fc2_membered,t3_subset,e2_3__binari_4,e1_3__binari_4,t119_zfmisc_1]), [interesting(0.8),file(binari_4,e3_3__binari_4),[file(binari_4,e3_3__binari_4)]]). fof(t84_funcop_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( r1_tarski(k2_zfmisc_1(k2_relat_1(B),k2_relat_1(C)),k1_relat_1(A)) => k1_relat_1(k3_funcop_1(A,B,C)) = k3_xboole_0(k1_relat_1(B),k1_relat_1(C)) ) ) ) ) ), file(funcop_1,t84_funcop_1), [interesting(0.9),axiom,file(funcop_1,t84_funcop_1)]). fof(e2_3_1__binari_4,plain,( k1_relat_1(k1_finseqop(k1_numbers,k1_numbers,k1_numbers,k33_binop_2,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4))) = k3_xboole_0(k1_relat_1(k5_euclid(c1_3__binari_4)),k1_relat_1(k5_euclid(c1_3__binari_4))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_margrel1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,antisymmetry_r2_hidden,existence_m1_finseq_2,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_margrel1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t2_boole,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_euclid,dt_k1_zfmisc_1,dt_k4_euclid,dt_k5_numbers,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,redefinition_k1_finseqop,redefinition_k5_euclid,dt_k1_finseqop,dt_k1_numbers,dt_k1_relat_1,dt_k2_relat_1,dt_k2_zfmisc_1,dt_k33_binop_2,dt_k3_funcop_1,dt_k3_xboole_0,dt_k5_euclid,dt_c1_3__binari_4,fc2_membered,t3_subset,e3_3__binari_4,t84_funcop_1]), [interesting(0.65),file(binari_4,e2_3_1__binari_4),[file(binari_4,e2_3_1__binari_4)]]). fof(e3_3_1__binari_4,plain,( k3_xboole_0(k1_relat_1(k5_euclid(c1_3__binari_4)),k1_relat_1(k5_euclid(c1_3__binari_4))) = k1_relat_1(k4_finseqop(k1_numbers,c1_3__binari_4,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,rc1_margrel1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_finseq_2,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_margrel1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_boole,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,redefinition_m2_finseq_1,redefinition_m2_finseq_2,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_finseq_2,dt_k4_finseq_2,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,existence_m2_subset_1,redefinition_k4_finseqop,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_relat_1,dt_k3_xboole_0,dt_k4_euclid,dt_k4_finseqop,dt_k5_euclid,dt_k5_numbers,dt_m2_subset_1,dt_c1_3__binari_4,fc2_membered,spc0_boole,spc0_numerals,d4_euclid]), [interesting(0.65),file(binari_4,e3_3_1__binari_4),[file(binari_4,e3_3_1__binari_4)]]). fof(e4_3_1__binari_4,plain,( k1_relat_1(k4_finseqop(k1_numbers,c1_3__binari_4,0)) = k2_finseq_1(c1_3__binari_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,rc1_margrel1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k4_finseq_2,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,redefinition_k2_finseq_1,redefinition_k4_finseq_1,redefinition_k4_finseqop,dt_k1_numbers,dt_k1_relat_1,dt_k2_finseq_1,dt_k2_finseq_2,dt_k4_finseq_1,dt_k4_finseqop,dt_c1_3__binari_4,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc2_membered,spc0_boole,spc0_numerals,t68_finseq_2]), [interesting(0.65),file(binari_4,e4_3_1__binari_4),[file(binari_4,e4_3_1__binari_4)]]). fof(e4_3__binari_4,plain,( k1_relat_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4))) = k2_finseq_1(c1_3__binari_4) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_3__binari_4])],[e1_3_1__binari_4,e2_3_1__binari_4,e3_3_1__binari_4,e4_3_1__binari_4]), [interesting(0.8),file(binari_4,e4_3__binari_4),[file(binari_4,e4_3__binari_4)]]). fof(t26_rvsum_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ! [C] : ( m2_finseq_1(C,k1_numbers) => ( r2_hidden(A,k4_finseq_1(k3_rvsum_1(B,C))) => k2_seq_1(k5_numbers,k1_numbers,k3_rvsum_1(B,C),A) = k9_binop_2(k2_seq_1(k5_numbers,k1_numbers,B,A),k2_seq_1(k5_numbers,k1_numbers,C,A)) ) ) ) ) ), file(rvsum_1,t26_rvsum_1), [interesting(0.9),axiom,file(rvsum_1,t26_rvsum_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__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(e3_3_3__binari_4,plain,( k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),c1_3_3__binari_4) = k1_nat_1(0,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3_3__binari_4,dt_c1_3__binari_4,e1_3_3__binari_4])],[rc1_margrel1,reflexivity_r1_tarski,existence_m1_finseq_2,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m1_finseq_2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_margrel1,fc3_xreal_0,fc6_int_1,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_relset_1,redefinition_m2_finseq_2,redefinition_m2_relset_1,dt_k1_euclid,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k4_euclid,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc5_membered,spc6_arithm,t1_arithm,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_rvsum_1,commutativity_k7_euclid,commutativity_k9_binop_2,antisymmetry_r2_hidden,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k2_seq_1,redefinition_k4_finseq_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_k7_euclid,redefinition_k9_binop_2,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_finseq_1,dt_k2_seq_1,dt_k2_xcmplx_0,dt_k3_rvsum_1,dt_k4_finseq_1,dt_k5_euclid,dt_k5_numbers,dt_k7_euclid,dt_k9_binop_2,dt_m2_finseq_1,dt_m2_subset_1,dt_c1_3__binari_4,dt_c1_3_3__binari_4,fc2_membered,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,t1_subset,t7_boole,spc0_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e2_3_3__binari_4,e4_3__binari_4,e5_3__binari_4,e1_3_3__binari_4,t26_rvsum_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r1_r1_r2]), [interesting(0.65),file(binari_4,e3_3_3__binari_4),[file(binari_4,e3_3_3__binari_4)]]). fof(e4_3_3__binari_4,plain,( k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),c1_3_3__binari_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3_3__binari_4,dt_c1_3__binari_4,e1_3_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,reflexivity_r1_tarski,existence_m2_relset_1,redefinition_m2_relset_1,dt_m2_relset_1,rc1_margrel1,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_finseq_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_margrel1,fc1_nat_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_membered,fc6_int_1,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k3_rvsum_1,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_euclid,dt_k1_numbers,dt_k3_rvsum_1,dt_k4_euclid,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,spc6_arithm,t1_arithm,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k7_euclid,redefinition_k1_nat_1,redefinition_k5_euclid,redefinition_k7_euclid,dt_k1_funct_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k5_euclid,dt_k7_euclid,dt_c1_3__binari_4,dt_c1_3_3__binari_4,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_boole,spc1_boole,spc2_boole,spc0_numerals,spc1_numerals,spc2_numerals,e3_3_3__binari_4,e2_3_3__binari_4,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r0_r0_r0]), [interesting(0.65),file(binari_4,e4_3_3__binari_4),[file(binari_4,e4_3_3__binari_4)]]). fof(i3_3_3__binari_4,theorem,( $true ), introduced(tautology,[file(binari_4,i3_3_3__binari_4)]), [interesting(0.65),trivial,file(binari_4,i3_3_3__binari_4)]). fof(i2_3_3__binari_4,plain,( k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),c1_3_3__binari_4) ), inference(conclusion,[status(thm),assumptions([dt_c1_3_3__binari_4,dt_c1_3__binari_4,e1_3_3__binari_4])],[e4_3_3__binari_4,i3_3_3__binari_4]), [interesting(0.65),file(binari_4,i2_3_3__binari_4),[file(binari_4,i2_3_3__binari_4)]]). fof(i1_3_3__binari_4,plain, ( r2_hidden(c1_3_3__binari_4,k1_relat_1(k5_euclid(c1_3__binari_4))) => k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),c1_3_3__binari_4) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_3_3__binari_4,dt_c1_3__binari_4]),discharge_asm(discharge,[e1_3_3__binari_4])],[e1_3_3__binari_4,i2_3_3__binari_4]), [interesting(0.65),file(binari_4,i1_3_3__binari_4),[file(binari_4,i1_3_3__binari_4)]]). fof(i1_3_3_tmp__binari_4,plain, ( m2_subset_1(c1_3_3__binari_4,k1_numbers,k5_numbers) => ( r2_hidden(c1_3_3__binari_4,k1_relat_1(k5_euclid(c1_3__binari_4))) => k1_funct_1(k5_euclid(c1_3__binari_4),c1_3_3__binari_4) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),c1_3_3__binari_4) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_3__binari_4]),discharge_asm(discharge,[dt_c1_3_3__binari_4])],[dt_c1_3_3__binari_4,i1_3_3__binari_4]), [interesting(0.8),e6_3__binari_4]). fof(e6_3__binari_4,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r2_hidden(A,k1_relat_1(k5_euclid(c1_3__binari_4))) => k1_funct_1(k5_euclid(c1_3__binari_4),A) = k1_funct_1(k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)),A) ) ) ), inference(let,[status(thm),assumptions([dt_c1_3__binari_4])],[i1_3_3_tmp__binari_4,dh_c1_3_3__binari_4]), [interesting(0.8),file(binari_4,e6_3__binari_4),[file(binari_4,e6_3__binari_4)]]). fof(t17_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ( k4_finseq_1(A) = k4_finseq_1(B) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( r2_hidden(C,k4_finseq_1(A)) => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) => A = B ) ) ) ), file(finseq_1,t17_finseq_1), [interesting(0.9),axiom,file(finseq_1,t17_finseq_1)]). fof(e7_3__binari_4,plain,( k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)) = k5_euclid(c1_3__binari_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_3__binari_4])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m2_relset_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_margrel1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_margrel1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,commutativity_k3_rvsum_1,existence_m1_subset_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k1_euclid,dt_k1_finseq_1,dt_k1_zfmisc_1,dt_k3_rvsum_1,dt_k4_euclid,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_finseq_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k7_euclid,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k2_finseq_1,redefinition_k4_finseq_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_k7_euclid,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_finseq_1,dt_k4_finseq_1,dt_k5_euclid,dt_k5_numbers,dt_k7_euclid,dt_m2_subset_1,dt_c1_3__binari_4,fc2_membered,t1_subset,t7_boole,e6_3__binari_4,e4_3__binari_4,e5_3__binari_4,t17_finseq_1]), [interesting(0.8),file(binari_4,e7_3__binari_4),[file(binari_4,e7_3__binari_4)]]). fof(i2_3__binari_4,theorem,( $true ), introduced(tautology,[file(binari_4,i2_3__binari_4)]), [interesting(0.8),trivial,file(binari_4,i2_3__binari_4)]). fof(i1_3__binari_4,plain,( k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)) = k5_euclid(c1_3__binari_4) ), inference(conclusion,[status(thm),assumptions([dt_c1_3__binari_4])],[e7_3__binari_4,i2_3__binari_4]), [interesting(0.8),file(binari_4,i1_3__binari_4),[file(binari_4,i1_3__binari_4)]]). fof(i1_3_tmp__binari_4,plain, ( m2_subset_1(c1_3__binari_4,k1_numbers,k5_numbers) => k7_euclid(c1_3__binari_4,k5_euclid(c1_3__binari_4),k5_euclid(c1_3__binari_4)) = k5_euclid(c1_3__binari_4) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__binari_4])],[dt_c1_3__binari_4,i1_3__binari_4]), [interesting(1),t3_binari_4]). fof(t3_binari_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k7_euclid(A,k5_euclid(A),k5_euclid(A)) = k5_euclid(A) ) ), inference(let,[status(thm),assumptions([])],[i1_3_tmp__binari_4,dh_c1_3__binari_4]), [interesting(1),file(binari_4,t3_binari_4),[file(binari_4,t3_binari_4)]]).