% Mizar ND problem: t7_binari_3,binari_3,198,54 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(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_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_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(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(rc2_margrel1,theorem,( ? [A] : v2_margrel1(A) ), file(margrel1,rc2_margrel1), [interesting(0.9),axiom,file(margrel1,rc2_margrel1)]). 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(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_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_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_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_finseq_2,axiom,( $true ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). 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_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(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_margrel1,theorem,( ! [A] : ( m1_subset_1(A,k6_margrel1) => v2_margrel1(A) ) ), file(margrel1,cc1_margrel1), [interesting(0.9),axiom,file(margrel1,cc1_margrel1)]). 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_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(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_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_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(cc4_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(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(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_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(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_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k4_finseq_2,axiom,( ! [A,B] : ( v4_ordinal2(A) => m1_finseq_2(k4_finseq_2(A,B),B) ) ), file(finseq_2,k4_finseq_2), [interesting(0.9),axiom,file(finseq_2,k4_finseq_2)]). fof(dt_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_k6_margrel1,axiom,( $true ), file(margrel1,k6_margrel1), [interesting(0.9),axiom,file(margrel1,k6_margrel1)]). fof(dt_k9_binarith,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) ) => m2_subset_1(k9_binarith(A,B),k1_numbers,k5_numbers) ) ), file(binarith,k9_binarith), [interesting(0.9),axiom,file(binarith,k9_binarith)]). 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(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(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(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(fc3_margrel1,theorem,( ~ v1_xboole_0(k6_margrel1) ), file(margrel1,fc3_margrel1), [interesting(0.9),axiom,file(margrel1,fc3_margrel1)]). 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(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(fc1_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), [interesting(0.9),axiom,file(nat_1,fc1_nat_1)]). fof(fc3_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), [interesting(0.9),axiom,file(nat_1,fc3_nat_1)]). fof(fc4_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), [interesting(0.9),axiom,file(nat_1,fc4_nat_1)]). fof(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_k23_binop_2,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k23_binop_2(B,A) ) ), file(binop_2,k23_binop_2), [interesting(0.9),axiom,file(binop_2,k23_binop_2)]). 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(redefinition_k23_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k23_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k23_binop_2), [interesting(0.9),axiom,file(binop_2,k23_binop_2)]). 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_k23_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k23_binop_2(A,B),k1_numbers,k5_numbers) ) ), file(binop_2,k23_binop_2), [interesting(0.9),axiom,file(binop_2,k23_binop_2)]). 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_binarith__e3_7__binari_3,theorem, ( ( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( A = k5_euclid(1) => k9_binarith(1,A) = 0 ) ) & ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( ! [C] : ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) => ( C = k5_euclid(B) => k9_binarith(B,C) = 0 ) ) => ! [D] : ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(k23_binop_2(B,1),k6_margrel1)) => ( D = k5_euclid(k23_binop_2(B,1)) => k9_binarith(k23_binop_2(B,1),D) = 0 ) ) ) ) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ! [E] : ( m2_finseq_2(E,k6_margrel1,k4_finseq_2(B,k6_margrel1)) => ( E = k5_euclid(B) => k9_binarith(B,E) = 0 ) ) ) ), file(binari_3,s1_binarith__e3_7__binari_3), [interesting(0.9),axiom,file(binari_3,s1_binarith__e3_7__binari_3)]). fof(dh_c1_7_1__binari_3,definition, ( ( m2_finseq_2(c1_7_1__binari_3,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( c1_7_1__binari_3 = k5_euclid(1) => k9_binarith(1,c1_7_1__binari_3) = 0 ) ) => ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( A = k5_euclid(1) => k9_binarith(1,A) = 0 ) ) ), introduced(definition,[new_symbol(c1_7_1__binari_3),file(binari_3,c1_7_1__binari_3)]), [interesting(0.65),axiom,file(binari_3,c1_7_1__binari_3)]). fof(e1_7_1__binari_3,assumption,( c1_7_1__binari_3 = k5_euclid(1) ), introduced(assumption,[file(binari_3,e1_7_1__binari_3)]), [interesting(0.65),axiom,file(binari_3,e1_7_1__binari_3)]). 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_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(fc14_membered,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) ) ) ), file(membered,fc14_membered), [interesting(0.9),axiom,file(membered,fc14_membered)]). fof(fc15_membered,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) ) ) ), file(membered,fc15_membered), [interesting(0.9),axiom,file(membered,fc15_membered)]). fof(rc1_margrel1,theorem,( ? [A] : v1_margrel1(A) ), file(margrel1,rc1_margrel1), [interesting(0.9),axiom,file(margrel1,rc1_margrel1)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(existence_m1_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_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(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(fc12_membered,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => v1_membered(k2_tarski(A,B)) ) ), file(membered,fc12_membered), [interesting(0.9),axiom,file(membered,fc12_membered)]). fof(fc13_membered,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) ) ) ), file(membered,fc13_membered), [interesting(0.9),axiom,file(membered,fc13_membered)]). 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(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(commutativity_k2_tarski,theorem,( ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(existence_m1_finseq_2,axiom,( ! [A] : ? [B] : m1_finseq_2(B,A) ), file(finseq_2,m1_finseq_2), [interesting(0.9),axiom,file(finseq_2,m1_finseq_2)]). fof(existence_m1_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_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(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(dt_k2_tarski,axiom,( $true ), file(tarski,k2_tarski), [interesting(0.9),axiom,file(tarski,k2_tarski)]). fof(dt_k5_finseq_1,axiom,( $true ), file(finseq_1,k5_finseq_1), [interesting(0.9),axiom,file(finseq_1,k5_finseq_1)]). fof(fc16_membered,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v1_membered(k2_tarski(A,B)) & v2_membered(k2_tarski(A,B)) & v3_membered(k2_tarski(A,B)) & v4_membered(k2_tarski(A,B)) & v5_membered(k2_tarski(A,B)) ) ) ), file(membered,fc16_membered), [interesting(0.9),axiom,file(membered,fc16_membered)]). 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(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_k12_finseq_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k12_finseq_1(A,B) = k5_finseq_1(B) ) ), file(finseq_1,k12_finseq_1), [interesting(0.9),axiom,file(finseq_1,k12_finseq_1)]). fof(redefinition_k13_binarith,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k13_binarith(A,B) = k5_finseq_1(B) ) ), file(binarith,k13_binarith), [interesting(0.9),axiom,file(binarith,k13_binarith)]). fof(dt_k12_finseq_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_1(k12_finseq_1(A,B),A) ) ), file(finseq_1,k12_finseq_1), [interesting(0.9),axiom,file(finseq_1,k12_finseq_1)]). fof(dt_k13_binarith,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(1,A)) ) ), file(binarith,k13_binarith), [interesting(0.9),axiom,file(binarith,k13_binarith)]). fof(dt_k7_margrel1,axiom,( m1_subset_1(k7_margrel1,k6_margrel1) ), file(margrel1,k7_margrel1), [interesting(0.9),axiom,file(margrel1,k7_margrel1)]). fof(dt_c1_7_1__binari_3,assumption,( m2_finseq_2(c1_7_1__binari_3,k6_margrel1,k4_finseq_2(1,k6_margrel1)) ), introduced(assumption,[file(binari_3,c1_7_1__binari_3)]), [interesting(0.65),axiom,file(binari_3,c1_7_1__binari_3)]). fof(d12_margrel1,definition,( k6_margrel1 = k2_tarski(0,1) ), file(margrel1,d12_margrel1), [interesting(0.9),axiom,file(margrel1,d12_margrel1)]). fof(d13_margrel1,definition,( k7_margrel1 = 0 ), file(margrel1,d13_margrel1), [interesting(0.9),axiom,file(margrel1,d13_margrel1)]). 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(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_7_1_1__binari_3,plain,( c1_7_1__binari_3 = k4_finseqop(k1_numbers,1,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_1__binari_3,e1_7_1__binari_3])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_membered,fc15_membered,rc1_margrel1,commutativity_k2_tarski,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_k2_tarski,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc3_nat_1,fc12_membered,fc13_membered,fc16_membered,fc1_margrel1,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,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_k6_margrel1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc3_margrel1,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,d12_margrel1,existence_m2_subset_1,redefinition_k4_finseqop,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k4_euclid,dt_k4_finseqop,dt_k5_euclid,dt_k5_numbers,dt_m2_subset_1,dt_c1_7_1__binari_3,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_7_1__binari_3,d4_euclid]), [interesting(0.5),file(binari_3,e1_7_1_1__binari_3),[file(binari_3,e1_7_1_1__binari_3)]]). fof(redefinition_k9_finseq_1,definition,( ! [A] : k9_finseq_1(A) = k5_finseq_1(A) ), file(finseq_1,k9_finseq_1), [interesting(0.9),axiom,file(finseq_1,k9_finseq_1)]). fof(dt_k8_margrel1,axiom,( m1_subset_1(k8_margrel1,k6_margrel1) ), file(margrel1,k8_margrel1), [interesting(0.9),axiom,file(margrel1,k8_margrel1)]). fof(dt_k9_finseq_1,axiom,( ! [A] : ( v1_relat_1(k9_finseq_1(A)) & v1_funct_1(k9_finseq_1(A)) ) ), file(finseq_1,k9_finseq_1), [interesting(0.9),axiom,file(finseq_1,k9_finseq_1)]). fof(d14_margrel1,definition,( k8_margrel1 = 1 ), file(margrel1,d14_margrel1), [interesting(0.9),axiom,file(margrel1,d14_margrel1)]). fof(t73_finseq_2,theorem,( ! [A] : k2_finseq_2(1,A) = k9_finseq_1(A) ), file(finseq_2,t73_finseq_2), [interesting(0.9),axiom,file(finseq_2,t73_finseq_2)]). fof(t36_margrel1,theorem, ( k7_margrel1 = 0 & k8_margrel1 = 1 ), file(margrel1,t36_margrel1), [interesting(0.9),axiom,file(margrel1,t36_margrel1)]). fof(e2_7_1_1__binari_3,plain,( k4_finseqop(k1_numbers,1,0) = k13_binarith(k6_margrel1,k7_margrel1) ), inference(mizar_by,[status(thm),assumptions([])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,fc14_membered,fc15_membered,rc1_margrel1,antisymmetry_r2_hidden,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_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_tarski,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k2_tarski,dt_k4_finseq_2,dt_k5_finseq_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc3_nat_1,cc4_membered,fc16_membered,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,redefinition_k13_binarith,redefinition_k4_finseqop,redefinition_k9_finseq_1,dt_k13_binarith,dt_k1_numbers,dt_k2_finseq_2,dt_k4_finseqop,dt_k6_margrel1,dt_k7_margrel1,dt_k8_margrel1,dt_k9_finseq_1,fc2_membered,fc3_margrel1,d12_margrel1,d13_margrel1,d14_margrel1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t73_finseq_2,t36_margrel1]), [interesting(0.5),file(binari_3,e2_7_1_1__binari_3),[file(binari_3,e2_7_1_1__binari_3)]]). fof(e2_7_1__binari_3,plain,( c1_7_1__binari_3 = k13_binarith(k6_margrel1,k7_margrel1) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_7_1__binari_3,e1_7_1__binari_3])],[e1_7_1_1__binari_3,e2_7_1_1__binari_3]), [interesting(0.65),file(binari_3,e2_7_1__binari_3),[file(binari_3,e2_7_1__binari_3)]]). fof(t41_binarith,theorem,( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( A = k12_finseq_1(k6_margrel1,k7_margrel1) => k9_binarith(1,A) = 0 ) ) ), file(binarith,t41_binarith), [interesting(0.9),axiom,file(binarith,t41_binarith)]). fof(e3_7_1__binari_3,plain,( k9_binarith(1,c1_7_1__binari_3) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_1__binari_3,e1_7_1__binari_3])],[reflexivity_r1_tarski,existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_membered,fc15_membered,rc1_margrel1,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,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_membered,cc4_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_tarski,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_tarski,dt_k5_finseq_1,dt_k5_numbers,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc3_nat_1,fc16_membered,fc2_membered,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,existence_m2_finseq_2,redefinition_k12_finseq_1,redefinition_k13_binarith,redefinition_m2_finseq_2,dt_k12_finseq_1,dt_k13_binarith,dt_k4_finseq_2,dt_k6_margrel1,dt_k7_margrel1,dt_k9_binarith,dt_m2_finseq_2,dt_c1_7_1__binari_3,fc3_margrel1,d12_margrel1,d13_margrel1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e2_7_1__binari_3,t41_binarith]), [interesting(0.65),file(binari_3,e3_7_1__binari_3),[file(binari_3,e3_7_1__binari_3)]]). fof(i3_7_1__binari_3,theorem,( $true ), introduced(tautology,[file(binari_3,i3_7_1__binari_3)]), [interesting(0.65),trivial,file(binari_3,i3_7_1__binari_3)]). fof(i2_7_1__binari_3,plain,( k9_binarith(1,c1_7_1__binari_3) = 0 ), inference(conclusion,[status(thm),assumptions([dt_c1_7_1__binari_3,e1_7_1__binari_3])],[e3_7_1__binari_3,i3_7_1__binari_3]), [interesting(0.65),file(binari_3,i2_7_1__binari_3),[file(binari_3,i2_7_1__binari_3)]]). fof(i1_7_1__binari_3,plain, ( c1_7_1__binari_3 = k5_euclid(1) => k9_binarith(1,c1_7_1__binari_3) = 0 ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7_1__binari_3]),discharge_asm(discharge,[e1_7_1__binari_3])],[e1_7_1__binari_3,i2_7_1__binari_3]), [interesting(0.65),file(binari_3,i1_7_1__binari_3),[file(binari_3,i1_7_1__binari_3)]]). fof(i1_7_1_tmp__binari_3,plain, ( m2_finseq_2(c1_7_1__binari_3,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( c1_7_1__binari_3 = k5_euclid(1) => k9_binarith(1,c1_7_1__binari_3) = 0 ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_7_1__binari_3])],[dt_c1_7_1__binari_3,i1_7_1__binari_3]), [interesting(0.8),e1_7__binari_3]). fof(e1_7__binari_3,plain,( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(1,k6_margrel1)) => ( A = k5_euclid(1) => k9_binarith(1,A) = 0 ) ) ), inference(let,[status(thm),assumptions([])],[i1_7_1_tmp__binari_3,dh_c1_7_1__binari_3]), [interesting(0.8),file(binari_3,e1_7__binari_3),[file(binari_3,e1_7__binari_3)]]). fof(dh_c1_7_2__binari_3,definition, ( ( ( ~ v1_xboole_0(c1_7_2__binari_3) & m2_subset_1(c1_7_2__binari_3,k1_numbers,k5_numbers) ) => ( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(c1_7_2__binari_3,k6_margrel1)) => ( A = k5_euclid(c1_7_2__binari_3) => k9_binarith(c1_7_2__binari_3,A) = 0 ) ) => ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( A = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),A) = 0 ) ) ) ) => ! [B] : ( ( ~ v1_xboole_0(B) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( ! [C] : ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) => ( C = k5_euclid(B) => k9_binarith(B,C) = 0 ) ) => ! [C] : ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(k1_nat_1(B,1),k6_margrel1)) => ( C = k5_euclid(k1_nat_1(B,1)) => k9_binarith(k1_nat_1(B,1),C) = 0 ) ) ) ) ), introduced(definition,[new_symbol(c1_7_2__binari_3),file(binari_3,c1_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,c1_7_2__binari_3)]). fof(e1_7_2__binari_3,assumption,( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(c1_7_2__binari_3,k6_margrel1)) => ( A = k5_euclid(c1_7_2__binari_3) => k9_binarith(c1_7_2__binari_3,A) = 0 ) ) ), introduced(assumption,[file(binari_3,e1_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,e1_7_2__binari_3)]). fof(dh_c2_7_2__binari_3,definition, ( ( m2_finseq_2(c2_7_2__binari_3,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( c2_7_2__binari_3 = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),c2_7_2__binari_3) = 0 ) ) => ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( A = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),A) = 0 ) ) ), introduced(definition,[new_symbol(c2_7_2__binari_3),file(binari_3,c2_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,c2_7_2__binari_3)]). fof(e6_7_2__binari_3,assumption,( c2_7_2__binari_3 = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) ), introduced(assumption,[file(binari_3,e6_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,e6_7_2__binari_3)]). fof(dt_k1_cqc_lang,axiom,( $true ), file(cqc_lang,k1_cqc_lang), [interesting(0.9),axiom,file(cqc_lang,k1_cqc_lang)]). fof(dt_k3_power,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k3_power(A,B)) ) ), file(power,k3_power), [interesting(0.9),axiom,file(power,k3_power)]). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), [interesting(0.9),axiom,file(finseq_1,k7_finseq_1)]). fof(commutativity_k3_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(redefinition_k12_binarith,definition,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) & ~ v1_xboole_0(C) & m1_subset_1(D,k4_finseq_2(A,C)) & m1_subset_1(E,k4_finseq_2(B,C)) ) => k12_binarith(A,B,C,D,E) = k7_finseq_1(D,E) ) ), file(binarith,k12_binarith), [interesting(0.9),axiom,file(binarith,k12_binarith)]). fof(redefinition_k2_cqc_lang,definition,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => k2_cqc_lang(A,B,C,D,E) = k1_cqc_lang(B,C,D,E) ) ), file(cqc_lang,k2_cqc_lang), [interesting(0.9),axiom,file(cqc_lang,k2_cqc_lang)]). 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(redefinition_k3_series_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k3_series_1(A,B) = k3_power(A,B) ) ), file(series_1,k3_series_1), [interesting(0.9),axiom,file(series_1,k3_series_1)]). fof(dt_k12_binarith,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) & ~ v1_xboole_0(C) & m1_subset_1(D,k4_finseq_2(A,C)) & m1_subset_1(E,k4_finseq_2(B,C)) ) => m2_finseq_2(k12_binarith(A,B,C,D,E),C,k4_finseq_2(k23_binop_2(A,B),C)) ) ), file(binarith,k12_binarith), [interesting(0.9),axiom,file(binarith,k12_binarith)]). fof(dt_k2_cqc_lang,axiom,( ! [A,B,C,D,E] : ( ( ~ v1_xboole_0(A) & m1_subset_1(D,A) & m1_subset_1(E,A) ) => m1_subset_1(k2_cqc_lang(A,B,C,D,E),A) ) ), file(cqc_lang,k2_cqc_lang), [interesting(0.9),axiom,file(cqc_lang,k2_cqc_lang)]). 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(dt_k3_series_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k3_series_1(A,B),k1_numbers,k5_numbers) ) ), file(series_1,k3_series_1), [interesting(0.9),axiom,file(series_1,k3_series_1)]). fof(dt_c1_7_2__binari_3,assumption, ( ~ v1_xboole_0(c1_7_2__binari_3) & m2_subset_1(c1_7_2__binari_3,k1_numbers,k5_numbers) ), introduced(assumption,[file(binari_3,c1_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,c1_7_2__binari_3)]). fof(dt_c2_7_2__binari_3,assumption,( m2_finseq_2(c2_7_2__binari_3,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) ), introduced(assumption,[file(binari_3,c2_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,c2_7_2__binari_3)]). fof(de_c3_7_2__binari_3,definition,( c3_7_2__binari_3 = k5_euclid(c1_7_2__binari_3) ), introduced(definition,[new_symbol(c3_7_2__binari_3),file(binari_3,c3_7_2__binari_3)]), [interesting(0.65),axiom,file(binari_3,c3_7_2__binari_3)]). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), [interesting(0.9),axiom,file(card_1,k1_card_1)]). fof(redefinition_k3_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]). fof(t8_jordan2b,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k3_finseq_1(k5_euclid(A)) = A ) ), file(jordan2b,t8_jordan2b), [interesting(0.9),axiom,file(jordan2b,t8_jordan2b)]). fof(e4_7_2__binari_3,plain,( k3_finseq_1(k5_euclid(c1_7_2__binari_3)) = c1_7_2__binari_3 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[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_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,cc3_nat_1,fc1_margrel1,fc6_membered,rc1_membered,rc1_nat_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k1_card_1,dt_k1_euclid,dt_k1_zfmisc_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,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k3_finseq_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k3_finseq_1,dt_k5_euclid,dt_k5_numbers,dt_m2_subset_1,dt_c1_7_2__binari_3,fc2_membered,t8_jordan2b]), [interesting(0.65),file(binari_3,e4_7_2__binari_3),[file(binari_3,e4_7_2__binari_3)]]). fof(dt_k13_finseq_1,axiom,( $true ), file(finseq_1,k13_finseq_1), [interesting(0.9),axiom,file(finseq_1,k13_finseq_1)]). fof(t5_binari_3,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r2_hidden(k5_euclid(A),k13_finseq_1(k6_margrel1)) ) ), file(binari_3,t5_binari_3), [interesting(0.9),axiom,file(binari_3,t5_binari_3)]). fof(e2_7_2__binari_3,plain,( r2_hidden(k5_euclid(c1_7_2__binari_3),k13_finseq_1(k6_margrel1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[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,fc14_membered,fc15_membered,rc1_margrel1,reflexivity_r1_tarski,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_xboole_0,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,cc3_nat_1,fc12_membered,fc13_membered,fc16_membered,fc1_margrel1,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,commutativity_k2_tarski,existence_m1_subset_1,existence_m2_finseq_2,redefinition_m2_finseq_2,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_tarski,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_margrel1,cc1_nat_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,spc0_boole,spc1_boole,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k13_finseq_1,dt_k1_numbers,dt_k5_euclid,dt_k5_numbers,dt_k6_margrel1,dt_m2_subset_1,dt_c1_7_2__binari_3,fc2_membered,fc3_margrel1,t1_subset,t7_boole,d12_margrel1,t5_binari_3]), [interesting(0.65),file(binari_3,e2_7_2__binari_3),[file(binari_3,e2_7_2__binari_3)]]). fof(d11_finseq_1,definition,( ! [A,B] : ( B = k13_finseq_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> m2_finseq_1(C,A) ) ) ), file(finseq_1,d11_finseq_1), [interesting(0.9),axiom,file(finseq_1,d11_finseq_1)]). fof(e3_7_2__binari_3,plain,( m2_finseq_1(k5_euclid(c1_7_2__binari_3),k6_margrel1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[reflexivity_r1_tarski,fc14_membered,fc15_membered,rc1_margrel1,existence_m1_finseq_2,existence_m1_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc16_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,t3_subset,t4_subset,t5_subset,commutativity_k2_tarski,existence_m1_finseq_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_k1_numbers,dt_k2_tarski,dt_k4_euclid,dt_k5_numbers,dt_m1_finseq_1,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_relset_1,dt_m2_subset_1,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,fc2_membered,spc0_boole,spc1_boole,t1_numerals,t2_subset,t6_boole,t8_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,antisymmetry_r2_hidden,existence_m2_finseq_1,redefinition_k5_euclid,redefinition_m2_finseq_1,dt_k13_finseq_1,dt_k5_euclid,dt_k6_margrel1,dt_m2_finseq_1,dt_c1_7_2__binari_3,fc3_margrel1,t1_subset,t7_boole,d12_margrel1,e2_7_2__binari_3,d11_finseq_1]), [interesting(0.65),file(binari_3,e3_7_2__binari_3),[file(binari_3,e3_7_2__binari_3)]]). fof(t110_finseq_2,theorem,( ! [A,B] : ( m2_finseq_1(B,A) => m1_subset_1(B,k4_finseq_2(k3_finseq_1(B),A)) ) ), file(finseq_2,t110_finseq_2), [interesting(0.9),axiom,file(finseq_2,t110_finseq_2)]). fof(e5_7_2__binari_3,plain,( m2_finseq_2(k5_euclid(c1_7_2__binari_3),k6_margrel1,k4_finseq_2(c1_7_2__binari_3,k6_margrel1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[reflexivity_r1_tarski,fc14_membered,fc15_membered,rc1_margrel1,antisymmetry_r2_hidden,existence_m1_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_nat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_tarski,existence_m1_finseq_1,existence_m1_finseq_2,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_euclid,dt_k1_numbers,dt_k2_tarski,dt_k4_euclid,dt_k5_numbers,dt_m1_finseq_1,dt_m1_finseq_2,dt_m2_relset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_nat_1,cc3_nat_1,fc16_membered,fc2_membered,rc2_margrel1,spc0_boole,spc1_boole,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_finseq_2,redefinition_k3_finseq_1,redefinition_k5_euclid,redefinition_m2_finseq_1,redefinition_m2_finseq_2,dt_k3_finseq_1,dt_k4_finseq_2,dt_k5_euclid,dt_k6_margrel1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,dt_c1_7_2__binari_3,cc1_margrel1,fc3_margrel1,d12_margrel1,e4_7_2__binari_3,e3_7_2__binari_3,t110_finseq_2]), [interesting(0.65),file(binari_3,e5_7_2__binari_3),[file(binari_3,e5_7_2__binari_3)]]). fof(dt_c3_7_2__binari_3,plain,( m2_finseq_2(c3_7_2__binari_3,k6_margrel1,k4_finseq_2(c1_7_2__binari_3,k6_margrel1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[reflexivity_r1_tarski,existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_membered,fc15_membered,rc1_margrel1,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,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_membered,cc4_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_tarski,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_euclid,dt_k1_numbers,dt_k2_tarski,dt_k4_euclid,dt_k5_numbers,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc3_nat_1,fc16_membered,fc2_membered,spc0_boole,spc1_boole,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,existence_m2_finseq_2,redefinition_k5_euclid,redefinition_m2_finseq_2,dt_k4_finseq_2,dt_k5_euclid,dt_k6_margrel1,dt_m2_finseq_2,dt_c1_7_2__binari_3,fc3_margrel1,d12_margrel1,de_c3_7_2__binari_3,e5_7_2__binari_3]), [interesting(0.65),file(binari_3,c3_7_2__binari_3),[file(binari_3,c3_7_2__binari_3)]]). 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(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(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(redefinition_k8_finseq_1,definition,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), [interesting(0.9),axiom,file(finseq_1,k8_finseq_1)]). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), [interesting(0.9),axiom,file(finseq_1,k8_finseq_1)]). fof(t4_binari_3,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k5_euclid(k1_nat_1(A,1)) = k8_finseq_1(k1_numbers,k5_euclid(A),k13_binarith(k1_numbers,0)) ) ), file(binari_3,t4_binari_3), [interesting(0.9),axiom,file(binari_3,t4_binari_3)]). fof(e1_7_2_1__binari_3,plain,( k9_binarith(k1_nat_1(c1_7_2__binari_3,1),c2_7_2__binari_3) = k9_binarith(k23_binop_2(c1_7_2__binari_3,1),k12_binarith(c1_7_2__binari_3,1,k6_margrel1,c3_7_2__binari_3,k13_binarith(k6_margrel1,k7_margrel1))) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7_2__binari_3,dt_c1_7_2__binari_3,e6_7_2__binari_3])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_membered,fc15_membered,rc1_margrel1,reflexivity_r1_tarski,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_2,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc3_nat_1,fc12_membered,fc13_membered,fc16_membered,fc1_margrel1,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,spc6_arithm,t1_arithm,t1_subset,t4_subset,t5_subset,commutativity_k2_tarski,commutativity_k2_xcmplx_0,existence_m1_finseq_1,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_tarski,dt_k2_xcmplx_0,dt_k4_euclid,dt_k4_finseq_2,dt_k5_finseq_1,dt_k5_ordinal2,dt_k7_finseq_1,dt_m1_finseq_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_finseq_2,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,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,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k23_binop_2,existence_m2_subset_1,redefinition_k12_binarith,redefinition_k13_binarith,redefinition_k1_nat_1,redefinition_k23_binop_2,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_k8_finseq_1,redefinition_m2_subset_1,dt_k12_binarith,dt_k13_binarith,dt_k1_nat_1,dt_k1_numbers,dt_k23_binop_2,dt_k5_euclid,dt_k5_numbers,dt_k6_margrel1,dt_k7_margrel1,dt_k8_finseq_1,dt_k8_margrel1,dt_k9_binarith,dt_m2_subset_1,dt_c1_7_2__binari_3,dt_c2_7_2__binari_3,dt_c3_7_2__binari_3,de_c3_7_2__binari_3,fc2_membered,fc3_margrel1,d12_margrel1,d13_margrel1,d14_margrel1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e6_7_2__binari_3,t4_binari_3,t36_margrel1]), [interesting(0.5),file(binari_3,e1_7_2_1__binari_3),[file(binari_3,e1_7_2_1__binari_3)]]). 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_r1_r2,theorem,( k2_xcmplx_0(1,1) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2)]). fof(rqRealAdd__k2_xcmplx_0__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(t46_binarith,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1)) => ! [C] : ( m1_subset_1(C,k6_margrel1) => k9_binarith(k23_binop_2(A,1),k12_binarith(A,1,k6_margrel1,B,k13_binarith(k6_margrel1,C))) = k23_binop_2(k9_binarith(A,B),k2_cqc_lang(k5_numbers,C,k7_margrel1,0,k3_series_1(2,A))) ) ) ) ), file(binarith,t46_binarith), [interesting(0.9),axiom,file(binarith,t46_binarith)]). fof(e2_7_2_1__binari_3,plain,( k9_binarith(k23_binop_2(c1_7_2__binari_3,1),k12_binarith(c1_7_2__binari_3,1,k6_margrel1,c3_7_2__binari_3,k13_binarith(k6_margrel1,k7_margrel1))) = k3_real_1(k9_binarith(c1_7_2__binari_3,c3_7_2__binari_3),k2_cqc_lang(k1_numbers,k7_margrel1,k7_margrel1,0,k3_series_1(2,c1_7_2__binari_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_euclid,dt_k4_euclid,dt_m1_finseq_1,dt_m2_relset_1,fc14_membered,fc15_membered,rc1_margrel1,commutativity_k2_tarski,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_finseq_2,existence_m2_finseq_1,redefinition_k5_euclid,redefinition_m2_finseq_1,dt_k1_cqc_lang,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_tarski,dt_k2_xcmplx_0,dt_k3_power,dt_k5_euclid,dt_k5_finseq_1,dt_k5_ordinal2,dt_k7_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,cc20_membered,cc2_membered,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc16_membered,fc1_margrel1,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,spc6_arithm,t1_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k23_binop_2,commutativity_k3_real_1,existence_m1_subset_1,existence_m2_finseq_2,existence_m2_subset_1,redefinition_k12_binarith,redefinition_k13_binarith,redefinition_k23_binop_2,redefinition_k2_cqc_lang,redefinition_k3_real_1,redefinition_k3_series_1,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k12_binarith,dt_k13_binarith,dt_k1_numbers,dt_k23_binop_2,dt_k2_cqc_lang,dt_k3_real_1,dt_k3_series_1,dt_k4_finseq_2,dt_k5_numbers,dt_k6_margrel1,dt_k7_margrel1,dt_k9_binarith,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,dt_c1_7_2__binari_3,dt_c3_7_2__binari_3,de_c3_7_2__binari_3,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,fc2_membered,fc3_margrel1,spc0_boole,spc1_boole,spc2_boole,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,d12_margrel1,d13_margrel1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,t46_binarith]), [interesting(0.5),file(binari_3,e2_7_2_1__binari_3),[file(binari_3,e2_7_2_1__binari_3)]]). fof(e3_7_2_1__binari_3,plain,( k3_real_1(k9_binarith(c1_7_2__binari_3,c3_7_2__binari_3),k2_cqc_lang(k1_numbers,k7_margrel1,k7_margrel1,0,k3_series_1(2,c1_7_2__binari_3))) = k3_real_1(0,k2_cqc_lang(k1_numbers,k7_margrel1,k7_margrel1,0,k3_series_1(2,c1_7_2__binari_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3,e1_7_2__binari_3])],[reflexivity_r1_tarski,existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_membered,fc15_membered,rc1_margrel1,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc1_margrel1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,spc6_arithm,t1_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_tarski,commutativity_k2_xcmplx_0,existence_m1_finseq_2,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_cqc_lang,dt_k1_euclid,dt_k2_tarski,dt_k2_xcmplx_0,dt_k3_power,dt_k4_euclid,dt_k5_numbers,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc3_nat_1,cc4_membered,fc16_membered,fc1_nat_1,fc3_nat_1,fc4_nat_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,spc1_boole,spc1_numerals,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,spc1_numerals,spc1_boole,commutativity_k3_real_1,existence_m2_finseq_2,redefinition_k2_cqc_lang,redefinition_k3_real_1,redefinition_k3_series_1,redefinition_k5_euclid,redefinition_m2_finseq_2,dt_k1_numbers,dt_k2_cqc_lang,dt_k3_real_1,dt_k3_series_1,dt_k4_finseq_2,dt_k5_euclid,dt_k6_margrel1,dt_k7_margrel1,dt_k9_binarith,dt_m2_finseq_2,dt_c1_7_2__binari_3,dt_c3_7_2__binari_3,de_c3_7_2__binari_3,fc2_membered,fc3_margrel1,d12_margrel1,d13_margrel1,spc0_numerals,spc2_numerals,spc0_boole,spc2_boole,e1_7_2__binari_3]), [interesting(0.5),file(binari_3,e3_7_2_1__binari_3),[file(binari_3,e3_7_2_1__binari_3)]]). fof(d1_cqc_lang,definition,( ! [A,B,C,D] : ( ( A = B => k1_cqc_lang(A,B,C,D) = C ) & ( A != B => k1_cqc_lang(A,B,C,D) = D ) ) ), file(cqc_lang,d1_cqc_lang), [interesting(0.9),axiom,file(cqc_lang,d1_cqc_lang)]). fof(e4_7_2_1__binari_3,plain,( k3_real_1(0,k2_cqc_lang(k1_numbers,k7_margrel1,k7_margrel1,0,k3_series_1(2,c1_7_2__binari_3))) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_2__binari_3])],[reflexivity_r1_tarski,fc14_membered,fc15_membered,rc1_margrel1,commutativity_k2_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_tarski,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc3_nat_1,cc6_membered,cc9_membered,fc12_membered,fc13_membered,fc16_membered,fc1_margrel1,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc5_membered,fc6_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,spc1_boole,spc1_numerals,spc6_arithm,t1_arithm,t1_subset,t3_subset,t4_subset,t5_subset,spc1_numerals,spc1_boole,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k2_xcmplx_0,dt_k3_power,dt_k5_numbers,dt_k6_margrel1,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc4_membered,fc3_margrel1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r2_r0_r2,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,d12_margrel1,commutativity_k3_real_1,redefinition_k2_cqc_lang,redefinition_k3_real_1,redefinition_k3_series_1,dt_k1_cqc_lang,dt_k1_numbers,dt_k2_cqc_lang,dt_k3_real_1,dt_k3_series_1,dt_k7_margrel1,dt_c1_7_2__binari_3,fc2_membered,d13_margrel1,spc0_numerals,spc2_numerals,spc0_boole,spc2_boole,d1_cqc_lang]), [interesting(0.5),file(binari_3,e4_7_2_1__binari_3),[file(binari_3,e4_7_2_1__binari_3)]]). fof(e7_7_2__binari_3,plain,( k9_binarith(k1_nat_1(c1_7_2__binari_3,1),c2_7_2__binari_3) = 0 ), inference(iterative_eq,[status(thm),assumptions([dt_c2_7_2__binari_3,e6_7_2__binari_3,e1_7_2__binari_3,dt_c1_7_2__binari_3])],[dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,redefinition_m2_finseq_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_finseq_2,dt_m2_finseq_1,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc3_membered,cc3_nat_1,cc6_membered,cc9_membered,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc5_membered,rc1_membered,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,commutativity_k2_xcmplx_0,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_cqc_lang,dt_k2_xcmplx_0,dt_k3_power,dt_k4_finseq_2,dt_k5_finseq_1,dt_k5_numbers,dt_k7_finseq_1,dt_m1_subset_1,dt_m2_finseq_2,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc1_margrel1,cc1_nat_1,cc2_nat_1,cc4_membered,commutativity_k1_nat_1,commutativity_k23_binop_2,commutativity_k3_real_1,redefinition_k12_binarith,redefinition_k13_binarith,redefinition_k1_nat_1,redefinition_k23_binop_2,redefinition_k2_cqc_lang,redefinition_k3_real_1,redefinition_k3_series_1,dt_k12_binarith,dt_k13_binarith,dt_k1_nat_1,dt_k1_numbers,dt_k23_binop_2,dt_k2_cqc_lang,dt_k3_real_1,dt_k3_series_1,dt_k6_margrel1,dt_k7_margrel1,dt_k9_binarith,dt_c1_7_2__binari_3,dt_c2_7_2__binari_3,dt_c3_7_2__binari_3,fc2_membered,fc3_margrel1,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_7_2_1__binari_3,e2_7_2_1__binari_3,e3_7_2_1__binari_3,e4_7_2_1__binari_3]), [interesting(0.65),file(binari_3,e7_7_2__binari_3),[file(binari_3,e7_7_2__binari_3)]]). fof(i5_7_2__binari_3,theorem,( $true ), introduced(tautology,[file(binari_3,i5_7_2__binari_3)]), [interesting(0.65),trivial,file(binari_3,i5_7_2__binari_3)]). fof(i4_7_2__binari_3,plain,( k9_binarith(k1_nat_1(c1_7_2__binari_3,1),c2_7_2__binari_3) = 0 ), inference(conclusion,[status(thm),assumptions([dt_c2_7_2__binari_3,e6_7_2__binari_3,e1_7_2__binari_3,dt_c1_7_2__binari_3])],[e7_7_2__binari_3,i5_7_2__binari_3]), [interesting(0.65),file(binari_3,i4_7_2__binari_3),[file(binari_3,i4_7_2__binari_3)]]). fof(i3_7_2__binari_3,plain, ( c2_7_2__binari_3 = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),c2_7_2__binari_3) = 0 ), inference(discharge_asm,[status(thm),assumptions([dt_c2_7_2__binari_3,e1_7_2__binari_3,dt_c1_7_2__binari_3]),discharge_asm(discharge,[e6_7_2__binari_3])],[e6_7_2__binari_3,i4_7_2__binari_3]), [interesting(0.65),file(binari_3,i3_7_2__binari_3),[file(binari_3,i3_7_2__binari_3)]]). fof(i3_7_2_tmp__binari_3,plain, ( m2_finseq_2(c2_7_2__binari_3,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( c2_7_2__binari_3 = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),c2_7_2__binari_3) = 0 ) ), inference(discharge_asm,[status(thm),assumptions([e1_7_2__binari_3,dt_c1_7_2__binari_3]),discharge_asm(discharge,[dt_c2_7_2__binari_3])],[dt_c2_7_2__binari_3,i3_7_2__binari_3]), [interesting(0.65),i2_7_2__binari_3]). fof(i2_7_2__binari_3,plain,( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( A = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),A) = 0 ) ) ), inference(let,[status(thm),assumptions([e1_7_2__binari_3,dt_c1_7_2__binari_3])],[i3_7_2_tmp__binari_3,dh_c2_7_2__binari_3]), [interesting(0.65),file(binari_3,i2_7_2__binari_3),[file(binari_3,i2_7_2__binari_3)]]). fof(i1_7_2__binari_3,plain, ( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(c1_7_2__binari_3,k6_margrel1)) => ( A = k5_euclid(c1_7_2__binari_3) => k9_binarith(c1_7_2__binari_3,A) = 0 ) ) => ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( A = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),A) = 0 ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7_2__binari_3]),discharge_asm(discharge,[e1_7_2__binari_3])],[e1_7_2__binari_3,i2_7_2__binari_3]), [interesting(0.65),file(binari_3,i1_7_2__binari_3),[file(binari_3,i1_7_2__binari_3)]]). fof(i1_7_2_tmp__binari_3,plain, ( ( ~ v1_xboole_0(c1_7_2__binari_3) & m2_subset_1(c1_7_2__binari_3,k1_numbers,k5_numbers) ) => ( ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(c1_7_2__binari_3,k6_margrel1)) => ( A = k5_euclid(c1_7_2__binari_3) => k9_binarith(c1_7_2__binari_3,A) = 0 ) ) => ! [A] : ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(k1_nat_1(c1_7_2__binari_3,1),k6_margrel1)) => ( A = k5_euclid(k1_nat_1(c1_7_2__binari_3,1)) => k9_binarith(k1_nat_1(c1_7_2__binari_3,1),A) = 0 ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_7_2__binari_3])],[dt_c1_7_2__binari_3,i1_7_2__binari_3]), [interesting(0.8),e2_7__binari_3]). fof(e2_7__binari_3,plain,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ( ! [B] : ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1)) => ( B = k5_euclid(A) => k9_binarith(A,B) = 0 ) ) => ! [B] : ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(k1_nat_1(A,1),k6_margrel1)) => ( B = k5_euclid(k1_nat_1(A,1)) => k9_binarith(k1_nat_1(A,1),B) = 0 ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_7_2_tmp__binari_3,dh_c1_7_2__binari_3]), [interesting(0.8),file(binari_3,e2_7__binari_3),[file(binari_3,e2_7__binari_3)]]). fof(e3_7__binari_3,plain,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1)) => ( B = k5_euclid(A) => k9_binarith(A,B) = 0 ) ) ) ), inference(mizar_from,[status(thm),assumptions([])],[dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,commutativity_k2_xcmplx_0,redefinition_m2_finseq_1,dt_k1_euclid,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k4_euclid,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_margrel1,cc1_membered,cc1_nat_1,cc20_membered,cc2_membered,cc2_nat_1,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc5_membered,rc1_membered,commutativity_k1_nat_1,commutativity_k23_binop_2,redefinition_k1_nat_1,redefinition_k23_binop_2,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k23_binop_2,dt_k4_finseq_2,dt_k5_euclid,dt_k5_numbers,dt_k6_margrel1,dt_k9_binarith,dt_m2_finseq_2,dt_m2_subset_1,cc15_membered,fc2_membered,fc3_margrel1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,s1_binarith__e3_7__binari_3,e1_7__binari_3,e2_7__binari_3]), [interesting(0.8),file(binari_3,e3_7__binari_3),[file(binari_3,e3_7__binari_3)]]). fof(i1_7__binari_3,theorem,( $true ), introduced(tautology,[file(binari_3,i1_7__binari_3)]), [interesting(0.8),trivial,file(binari_3,i1_7__binari_3)]). fof(t7_binari_3,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & m2_subset_1(A,k1_numbers,k5_numbers) ) => ! [B] : ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1)) => ( B = k5_euclid(A) => k9_binarith(A,B) = 0 ) ) ) ), inference(conclusion,[status(thm),assumptions([])],[dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,rc1_nat_1,rc2_margrel1,rc2_nat_1,rc3_nat_1,redefinition_m2_finseq_1,dt_k1_euclid,dt_k1_zfmisc_1,dt_k4_euclid,dt_k5_ordinal2,dt_m1_finseq_2,dt_m1_subset_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_margrel1,cc1_membered,cc1_nat_1,cc20_membered,cc2_membered,cc2_nat_1,cc3_membered,cc3_nat_1,cc4_membered,cc6_membered,cc9_membered,fc5_membered,rc1_membered,redefinition_k5_euclid,redefinition_k5_numbers,redefinition_m2_finseq_2,redefinition_m2_subset_1,dt_k1_numbers,dt_k4_finseq_2,dt_k5_euclid,dt_k5_numbers,dt_k6_margrel1,dt_k9_binarith,dt_m2_finseq_2,dt_m2_subset_1,cc15_membered,fc2_membered,fc3_margrel1,spc0_numerals,spc0_boole,e3_7__binari_3,i1_7__binari_3]), [interesting(1),file(binari_3,t7_binari_3),[file(binari_3,t7_binari_3)]]).