% Mizar ND problem: t79_prepower,prepower,1735,16 fof(dh_c1_87__prepower,definition, ( ( v1_xreal_0(c1_87__prepower) => ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_prepower(A) & m2_relset_1(A,k5_numbers,k1_numbers) & v4_seq_2(A) & k2_seq_2(A) = c1_87__prepower & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(A,B),c1_87__prepower) ) ) ) => ! [C] : ( v1_xreal_0(C) => ? [D] : ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & v1_prepower(D) & m2_relset_1(D,k5_numbers,k1_numbers) & v4_seq_2(D) & k2_seq_2(D) = C & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(D,E),C) ) ) ) ), introduced(definition,[new_symbol(c1_87__prepower),file(prepower,c1_87__prepower)]), [interesting(0.8),axiom,file(prepower,c1_87__prepower)]). 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(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc2_rat_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v1_rat_1(A) ) ) ), file(rat_1,cc2_rat_1), [interesting(0.9),axiom,file(rat_1,cc2_rat_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_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(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_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(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(fc1_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) => ( v1_xcmplx_0(k1_funct_1(A,B)) & v1_xreal_0(k1_funct_1(A,B)) ) ) ), file(seq_1,fc1_seq_1), [interesting(0.9),axiom,file(seq_1,fc1_seq_1)]). 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_prepower,theorem,( ? [A] : ( m1_relset_1(A,k5_numbers,k1_numbers) & v1_relat_1(A) & v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_seq_1(A) & v1_prepower(A) ) ), file(prepower,rc1_prepower), [interesting(0.9),axiom,file(prepower,rc1_prepower)]). fof(rc1_rat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_rat_1(A) ) ), file(rat_1,rc1_rat_1), [interesting(0.9),axiom,file(rat_1,rc1_rat_1)]). fof(rc1_seq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ), file(seq_1,rc1_seq_1), [interesting(0.9),axiom,file(seq_1,rc1_seq_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(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), [interesting(0.9),axiom,file(nat_1,rc3_nat_1)]). fof(rc3_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(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_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => v1_xreal_0(k1_seq_2(A)) ) ), file(seq_2,k1_seq_2), [interesting(0.9),axiom,file(seq_2,k1_seq_2)]). 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_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(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_rat_1,theorem,( ! [A] : ( v1_rat_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(rat_1,cc1_rat_1), [interesting(0.9),axiom,file(rat_1,cc1_rat_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(cc1_seq_1,theorem,( ! [A,B] : ( v2_membered(B) => ! [C] : ( m1_relset_1(C,A,B) => ( v1_funct_1(C) => ( v1_funct_1(C) & v1_seq_1(C) ) ) ) ) ), file(seq_1,cc1_seq_1), [interesting(0.9),axiom,file(seq_1,cc1_seq_1)]). fof(cc2_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(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(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc1_ordinal2,theorem, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]). fof(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(rc2_rat_1,theorem,( ? [A] : v1_rat_1(A) ), file(rat_1,rc2_rat_1), [interesting(0.9),axiom,file(rat_1,rc2_rat_1)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(redefinition_k10_prepower,definition,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_prepower(A) & m1_relset_1(A,k5_numbers,k1_numbers) & m1_subset_1(B,k5_numbers) ) => k10_prepower(A,B) = k1_funct_1(A,B) ) ), file(prepower,k10_prepower), [interesting(0.9),axiom,file(prepower,k10_prepower)]). fof(redefinition_k2_seq_2,definition,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => k2_seq_2(A) = k1_seq_2(A) ) ), file(seq_2,k2_seq_2), [interesting(0.9),axiom,file(seq_2,k2_seq_2)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). 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(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k10_prepower,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_prepower(A) & m1_relset_1(A,k5_numbers,k1_numbers) & m1_subset_1(B,k5_numbers) ) => v1_rat_1(k10_prepower(A,B)) ) ), file(prepower,k10_prepower), [interesting(0.9),axiom,file(prepower,k10_prepower)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k2_seq_2,axiom,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) ) => m1_subset_1(k2_seq_2(A),k1_numbers) ) ), file(seq_2,k2_seq_2), [interesting(0.9),axiom,file(seq_2,k2_seq_2)]). 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_m2_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_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_87__prepower,assumption,( v1_xreal_0(c1_87__prepower) ), introduced(assumption,[file(prepower,c1_87__prepower)]), [interesting(0.8),axiom,file(prepower,c1_87__prepower)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(existence_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(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(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_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(dh_c2_87__prepower,definition, ( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,B) = k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(B,1),c1_87__prepower)),k1_nat_1(B,1)) ) ) => ( v1_funct_1(c2_87__prepower) & v1_funct_2(c2_87__prepower,k5_numbers,k1_numbers) & m2_relset_1(c2_87__prepower,k5_numbers,k1_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,C) = k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(C,1),c1_87__prepower)),k1_nat_1(C,1)) ) ) ), introduced(definition,[new_symbol(c2_87__prepower),file(prepower,c2_87__prepower)]), [interesting(0.8),axiom,file(prepower,c2_87__prepower)]). fof(fc10_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(rat_1,fc10_rat_1), [interesting(0.9),axiom,file(rat_1,fc10_rat_1)]). 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_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ), file(rat_1,fc12_rat_1), [interesting(0.9),axiom,file(rat_1,fc12_rat_1)]). 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(fc13_rat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_rat_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(rat_1,fc13_rat_1), [interesting(0.9),axiom,file(rat_1,fc13_rat_1)]). fof(fc15_rat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_rat_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ), file(rat_1,fc15_rat_1), [interesting(0.9),axiom,file(rat_1,fc15_rat_1)]). fof(fc17_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & v1_rat_1(k7_xcmplx_0(A,B)) ) ) ), file(rat_1,fc17_rat_1), [interesting(0.9),axiom,file(rat_1,fc17_rat_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(fc1_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ), file(rat_1,fc1_rat_1), [interesting(0.9),axiom,file(rat_1,fc1_rat_1)]). fof(fc21_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc21_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc21_xreal_0)]). fof(fc22_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc22_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc22_xreal_0)]). fof(fc24_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc24_xreal_0)]). fof(fc27_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc27_xreal_0)]). fof(fc28_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc28_xreal_0)]). fof(fc29_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc29_xreal_0)]). fof(fc2_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k3_xcmplx_0(A,B)) & v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(nat_1,fc2_nat_1), [interesting(0.9),axiom,file(nat_1,fc2_nat_1)]). fof(fc2_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(rat_1,fc2_rat_1), [interesting(0.9),axiom,file(rat_1,fc2_rat_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(fc4_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(rat_1,fc4_rat_1), [interesting(0.9),axiom,file(rat_1,fc4_rat_1)]). fof(fc6_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ), file(rat_1,fc6_rat_1), [interesting(0.9),axiom,file(rat_1,fc6_rat_1)]). fof(fc7_rat_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_rat_1(k2_xcmplx_0(A,B)) ) ) ), file(rat_1,fc7_rat_1), [interesting(0.9),axiom,file(rat_1,fc7_rat_1)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc9_rat_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_rat_1(k3_xcmplx_0(A,B)) ) ) ), file(rat_1,fc9_rat_1), [interesting(0.9),axiom,file(rat_1,fc9_rat_1)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(commutativity_k2_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(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(fc23_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc23_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc23_xreal_0)]). fof(fc2_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_1), [interesting(0.9),axiom,file(int_1,fc2_int_1)]). fof(fc30_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc30_xreal_0)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(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_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(B,A)) & v1_xreal_0(k3_xcmplx_0(B,A)) & v1_int_1(k3_xcmplx_0(B,A)) ) ) ), file(int_1,fc7_int_1), [interesting(0.9),axiom,file(int_1,fc7_int_1)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). fof(commutativity_k1_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(commutativity_k3_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,B) = k3_xcmplx_0(B,A) ) ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(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_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(dt_k1_int_1,axiom,( ! [A] : ( v1_xreal_0(A) => v1_int_1(k1_int_1(A)) ) ), file(int_1,k1_int_1), [interesting(0.9),axiom,file(int_1,k1_int_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_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_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_0)]). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k7_xcmplx_0)]). 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(fc4_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc4_xreal_0)]). fof(fc6_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc6_xreal_0)]). 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_seq_1__e1_87__prepower,theorem,( ! [A] : ( v1_xreal_0(A) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,C) = k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(C,1),A)),k1_nat_1(C,1)) ) ) ) ), file(prepower,s1_seq_1__e1_87__prepower), [interesting(0.9),axiom,file(prepower,s1_seq_1__e1_87__prepower)]). fof(e1_87__prepower,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,B) = k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(B,1),c1_87__prepower)),k1_nat_1(B,1)) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_87__prepower])],[cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc17_rat_1,fc1_nat_1,fc1_rat_1,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc1_seq_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc3_xreal_0,cc4_int_1,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_int_1,fc1_ordinal2,fc23_xreal_0,fc2_int_1,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc6_int_1,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_int_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_k7_xcmplx_0,dt_m2_relset_1,dt_m2_subset_1,dt_c1_87__prepower,cc2_xreal_0,fc2_membered,fc4_xreal_0,fc6_xreal_0,spc1_numerals,spc1_boole,s1_seq_1__e1_87__prepower]), [interesting(0.8),file(prepower,e1_87__prepower),[file(prepower,e1_87__prepower)]]). fof(dt_c2_87__prepower,plain, ( v1_funct_1(c2_87__prepower) & v1_funct_2(c2_87__prepower,k5_numbers,k1_numbers) & m2_relset_1(c2_87__prepower,k5_numbers,k1_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_87__prepower])],[dh_c2_87__prepower,e1_87__prepower]), [interesting(0.8),file(prepower,c2_87__prepower),[file(prepower,c2_87__prepower)]]). fof(de_c3_87__prepower,definition,( c3_87__prepower = c2_87__prepower ), introduced(definition,[new_symbol(c3_87__prepower),file(prepower,c3_87__prepower)]), [interesting(0.8),axiom,file(prepower,c3_87__prepower)]). 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(dh_c1_87_1__prepower,definition, ( ( m2_subset_1(c1_87_1__prepower,k1_numbers,k5_numbers) => v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,c1_87_1__prepower)) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,A)) ) ), introduced(definition,[new_symbol(c1_87_1__prepower),file(prepower,c1_87_1__prepower)]), [interesting(0.65),axiom,file(prepower,c1_87_1__prepower)]). fof(spc4_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(A,k7_xcmplx_0(B,C)) = k7_xcmplx_0(k3_xcmplx_0(A,B),C) ) ), file(arithm,spc4_arithm), [interesting(0.9),axiom,file(arithm,spc4_arithm)]). fof(spc5_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ), file(arithm,spc5_arithm), [interesting(0.9),axiom,file(arithm,spc5_arithm)]). fof(spc6_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k2_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(A,k2_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(spc7_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k3_xcmplx_0(k3_xcmplx_0(A,B),C) = k3_xcmplx_0(A,k3_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(t3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(1,A) = A ) ), file(arithm,t3_arithm), [interesting(0.9),axiom,file(arithm,t3_arithm)]). fof(t6_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), [interesting(0.9),axiom,file(arithm,t6_arithm)]). fof(dt_c1_87_1__prepower,assumption,( m2_subset_1(c1_87_1__prepower,k1_numbers,k5_numbers) ), introduced(assumption,[file(prepower,c1_87_1__prepower)]), [interesting(0.65),axiom,file(prepower,c1_87_1__prepower)]). fof(rqRealDiv__k7_xcmplx_0__r1_r1_r1,theorem,( k7_xcmplx_0(1,1) = 1 ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1)]). fof(e2_87__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,A) = k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(A,1),c1_87__prepower)),k1_nat_1(A,1)) ) ), inference(consider,[status(thm),assumptions([dt_c1_87__prepower])],[dh_c2_87__prepower,e1_87__prepower]), [interesting(0.8),file(prepower,e2_87__prepower),[file(prepower,e2_87__prepower)]]). fof(rqRealMult__k3_xcmplx_0__r1_r1_r1,theorem,( k3_xcmplx_0(1,1) = 1 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r1_r1)]). fof(e1_87_1__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,c1_87_1__prepower) = k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_1__prepower,1),c1_87__prepower)),k1_nat_1(c1_87_1__prepower,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_1__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc17_rat_1,fc1_nat_1,fc1_rat_1,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc1_seq_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_int_1,fc1_ordinal2,fc23_xreal_0,fc2_int_1,fc30_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc6_int_1,fc6_xreal_0,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_int_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_87__prepower,dt_c1_87_1__prepower,dt_c2_87__prepower,fc2_membered,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc1_numerals,spc1_boole,e2_87__prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e1_87_1__prepower),[file(prepower,e1_87_1__prepower)]]). fof(e2_87_1__prepower,plain,( v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,c1_87_1__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_1__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_nat_1,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc3_nat_1,fc4_nat_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_seq_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc10_rat_1,fc12_rat_1,fc13_rat_1,fc15_rat_1,fc1_int_1,fc1_ordinal2,fc23_xreal_0,fc2_int_1,fc2_rat_1,fc30_xreal_0,fc3_xreal_0,fc4_rat_1,fc4_xreal_0,fc5_membered,fc6_int_1,fc6_rat_1,fc6_xreal_0,fc7_int_1,fc7_rat_1,fc8_xreal_0,fc9_rat_1,rc1_int_1,rc1_rat_1,rc1_xreal_0,rc2_int_1,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,dt_k1_int_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_1__prepower,dt_c2_87__prepower,cc1_rat_1,fc17_rat_1,fc1_rat_1,fc2_membered,rc2_rat_1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc1_numerals,spc1_boole,e1_87_1__prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e2_87_1__prepower),[file(prepower,e2_87_1__prepower)]]). fof(i2_87_1__prepower,theorem,( $true ), introduced(tautology,[file(prepower,i2_87_1__prepower)]), [interesting(0.65),trivial,file(prepower,i2_87_1__prepower)]). fof(i1_87_1__prepower,plain,( v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,c1_87_1__prepower)) ), inference(conclusion,[status(thm),assumptions([dt_c1_87_1__prepower,dt_c1_87__prepower])],[e2_87_1__prepower,i2_87_1__prepower]), [interesting(0.65),file(prepower,i1_87_1__prepower),[file(prepower,i1_87_1__prepower)]]). fof(i1_87_1_tmp__prepower,plain, ( m2_subset_1(c1_87_1__prepower,k1_numbers,k5_numbers) => v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,c1_87_1__prepower)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_87__prepower]),discharge_asm(discharge,[dt_c1_87_1__prepower])],[dt_c1_87_1__prepower,i1_87_1__prepower]), [interesting(0.8),e3_87__prepower]). fof(e3_87__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,c2_87__prepower,A)) ) ), inference(let,[status(thm),assumptions([dt_c1_87__prepower])],[i1_87_1_tmp__prepower,dh_c1_87_1__prepower]), [interesting(0.8),file(prepower,e3_87__prepower),[file(prepower,e3_87__prepower)]]). fof(d6_prepower,definition,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v1_prepower(A) <=> ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => v1_rat_1(k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ), file(prepower,d6_prepower), [interesting(0.9),axiom,file(prepower,d6_prepower)]). fof(e4_87__prepower,plain, ( v1_funct_1(c2_87__prepower) & v1_funct_2(c2_87__prepower,k5_numbers,k1_numbers) & v1_prepower(c2_87__prepower) & m2_relset_1(c2_87__prepower,k5_numbers,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_seq_1,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_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc1_rat_1,rc1_xreal_0,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_seq_1,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c2_87__prepower,cc1_rat_1,fc2_membered,rc2_rat_1,e3_87__prepower,d6_prepower]), [interesting(0.8),file(prepower,e4_87__prepower),[file(prepower,e4_87__prepower)]]). fof(dt_c3_87__prepower,plain, ( v1_funct_1(c3_87__prepower) & v1_funct_2(c3_87__prepower,k5_numbers,k1_numbers) & v1_prepower(c3_87__prepower) & m2_relset_1(c3_87__prepower,k5_numbers,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_rat_1,rc2_rat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_relset_1,dt_c2_87__prepower,fc2_membered,de_c3_87__prepower,e4_87__prepower]), [interesting(0.8),file(prepower,c3_87__prepower),[file(prepower,c3_87__prepower)]]). fof(cc4_seqm_3,theorem,( ! [A] : ( m1_relset_1(A,k5_numbers,k1_numbers) => ( ( v1_funct_1(A) & v3_seqm_3(A) & v4_seqm_3(A) ) => ( v1_funct_1(A) & v1_seq_1(A) & v5_seqm_3(A) ) ) ) ), file(seqm_3,cc4_seqm_3), [interesting(0.9),axiom,file(seqm_3,cc4_seqm_3)]). fof(cc3_seqm_3,theorem,( ! [A] : ( m1_relset_1(A,k5_numbers,k1_numbers) => ( ( v1_funct_1(A) & v5_seqm_3(A) ) => ( v1_funct_1(A) & v1_seq_1(A) & v3_seqm_3(A) & v4_seqm_3(A) ) ) ) ), file(seqm_3,cc3_seqm_3), [interesting(0.9),axiom,file(seqm_3,cc3_seqm_3)]). fof(fc4_ordinal2,theorem,( ! [A,B] : ( v3_ordinal1(B) => ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) & v1_ordinal2(k2_funcop_1(A,B)) ) ) ), file(ordinal2,fc4_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc4_ordinal2)]). fof(fc4_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k4_seq_1(A,B)) & v1_funct_1(k4_seq_1(A,B)) & v1_seq_1(k4_seq_1(A,B)) ) ) ), file(seq_1,fc4_seq_1), [interesting(0.9),axiom,file(seq_1,fc4_seq_1)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(dt_k2_funcop_1,axiom,( $true ), file(funcop_1,k2_funcop_1), [interesting(0.9),axiom,file(funcop_1,k2_funcop_1)]). fof(dt_k4_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k4_seq_1(A,B)) & v1_funct_1(k4_seq_1(A,B)) ) ) ), file(seq_1,k4_seq_1), [interesting(0.9),axiom,file(seq_1,k4_seq_1)]). fof(de_c4_87__prepower,definition,( c4_87__prepower = c1_87__prepower ), introduced(definition,[new_symbol(c4_87__prepower),file(prepower,c4_87__prepower)]), [interesting(0.8),axiom,file(prepower,c4_87__prepower)]). fof(d1_xreal_0,definition,( ! [A] : ( v1_xreal_0(A) <=> r2_hidden(A,k1_numbers) ) ), file(xreal_0,d1_xreal_0), [interesting(0.9),axiom,file(xreal_0,d1_xreal_0)]). fof(e5_87__prepower,plain,( m1_subset_1(c1_87__prepower,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc1_xreal_0,cc2_rat_1,cc3_int_1,cc3_nat_1,cc4_int_1,rc1_int_1,rc1_nat_1,rc1_rat_1,rc2_int_1,rc2_rat_1,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_membered,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t8_boole,cc10_membered,cc11_membered,cc15_membered,cc4_membered,cc7_xreal_0,rc1_xreal_0,t2_subset,t6_boole,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_numbers,dt_m1_subset_1,dt_c1_87__prepower,cc2_xreal_0,fc2_membered,t1_subset,t7_boole,d1_xreal_0]), [interesting(0.8),file(prepower,e5_87__prepower),[file(prepower,e5_87__prepower)]]). fof(dt_c4_87__prepower,plain,( m1_subset_1(c4_87__prepower,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc1_xreal_0,cc2_rat_1,cc3_int_1,cc3_nat_1,cc4_int_1,rc1_int_1,rc1_nat_1,rc1_rat_1,rc2_int_1,rc2_rat_1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc6_membered,rc1_membered,rc1_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,cc10_membered,cc11_membered,cc15_membered,cc2_xreal_0,cc4_membered,cc7_xreal_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,dt_k1_numbers,dt_m1_subset_1,dt_c1_87__prepower,fc2_membered,de_c4_87__prepower,e5_87__prepower]), [interesting(0.8),file(prepower,c4_87__prepower),[file(prepower,c4_87__prepower)]]). fof(fc2_seqm_3,theorem,( ! [A,B] : ( v1_relat_1(k2_funcop_1(A,B)) & v1_funct_1(k2_funcop_1(A,B)) & v5_seqm_3(k2_funcop_1(A,B)) ) ), file(seqm_3,fc2_seqm_3), [interesting(0.9),axiom,file(seqm_3,fc2_seqm_3)]). fof(redefinition_k10_seq_1,definition,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k10_seq_1(A,B) = k4_seq_1(A,B) ) ), file(seq_1,k10_seq_1), [interesting(0.9),axiom,file(seq_1,k10_seq_1)]). fof(dt_k10_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_funct_1(k10_seq_1(A,B)) & v1_funct_2(k10_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k10_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k10_seq_1), [interesting(0.9),axiom,file(seq_1,k10_seq_1)]). fof(de_c5_87__prepower,definition,( c5_87__prepower = k2_funcop_1(k5_numbers,c4_87__prepower) ), introduced(definition,[new_symbol(c5_87__prepower),file(prepower,c5_87__prepower)]), [interesting(0.8),axiom,file(prepower,c5_87__prepower)]). fof(t57_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(C,B) => ( v1_funct_1(k2_funcop_1(A,C)) & v1_funct_2(k2_funcop_1(A,C),A,B) & m2_relset_1(k2_funcop_1(A,C),A,B) ) ) ), file(funcop_1,t57_funcop_1), [interesting(0.9),axiom,file(funcop_1,t57_funcop_1)]). fof(e6_87__prepower,plain, ( v1_funct_1(k2_funcop_1(k5_numbers,c4_87__prepower)) & v1_funct_2(k2_funcop_1(k5_numbers,c4_87__prepower),k5_numbers,k1_numbers) & m2_relset_1(k2_funcop_1(k5_numbers,c4_87__prepower),k5_numbers,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_rat_1,rc2_rat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_seqm_3,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_c1_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc3_seqm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k1_numbers,dt_k2_funcop_1,dt_k5_numbers,dt_m2_relset_1,dt_c4_87__prepower,de_c4_87__prepower,fc2_membered,fc2_seqm_3,t1_subset,t7_boole,t57_funcop_1]), [interesting(0.8),file(prepower,e6_87__prepower),[file(prepower,e6_87__prepower)]]). fof(dt_c5_87__prepower,plain, ( v1_funct_1(c5_87__prepower) & v1_funct_2(c5_87__prepower,k5_numbers,k1_numbers) & m2_relset_1(c5_87__prepower,k5_numbers,k1_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_rat_1,rc2_rat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_seqm_3,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_c1_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc3_seqm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k1_numbers,dt_k2_funcop_1,dt_k5_numbers,dt_m2_relset_1,dt_c4_87__prepower,de_c4_87__prepower,fc2_membered,fc2_seqm_3,de_c5_87__prepower,e6_87__prepower]), [interesting(0.8),file(prepower,c5_87__prepower),[file(prepower,c5_87__prepower)]]). fof(dh_c6_87__prepower,definition, ( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_real_1(1,k1_nat_1(B,1)) ) ) => ( v1_funct_1(c6_87__prepower) & v1_funct_2(c6_87__prepower,k5_numbers,k1_numbers) & m2_relset_1(c6_87__prepower,k5_numbers,k1_numbers) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c6_87__prepower,C) = k6_real_1(1,k1_nat_1(C,1)) ) ) ), introduced(definition,[new_symbol(c6_87__prepower),file(prepower,c6_87__prepower)]), [interesting(0.8),axiom,file(prepower,c6_87__prepower)]). fof(redefinition_k6_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) ), file(real_1,k6_real_1), [interesting(0.9),axiom,file(real_1,k6_real_1)]). fof(dt_k6_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k6_real_1(A,B),k1_numbers) ) ), file(real_1,k6_real_1), [interesting(0.9),axiom,file(real_1,k6_real_1)]). fof(s1_seq_1__e8_87__prepower,theorem,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_real_1(1,k1_nat_1(B,1)) ) ) ), file(prepower,s1_seq_1__e8_87__prepower), [interesting(0.9),axiom,file(prepower,s1_seq_1__e8_87__prepower)]). fof(e8_87__prepower,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,B) = k6_real_1(1,k1_nat_1(B,1)) ) ) ), inference(mizar_from,[status(thm),assumptions([])],[cc1_rat_1,fc10_rat_1,fc13_rat_1,fc17_rat_1,fc2_rat_1,fc4_rat_1,fc7_rat_1,rc1_rat_1,rc2_rat_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,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,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc30_xreal_0,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc6_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k7_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,commutativity_k1_nat_1,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k5_numbers,dt_k6_real_1,dt_m2_relset_1,dt_m2_subset_1,fc2_membered,spc1_numerals,spc1_boole,s1_seq_1__e8_87__prepower]), [interesting(0.8),file(prepower,e8_87__prepower),[file(prepower,e8_87__prepower)]]). fof(dt_c6_87__prepower,plain, ( v1_funct_1(c6_87__prepower) & v1_funct_2(c6_87__prepower,k5_numbers,k1_numbers) & m2_relset_1(c6_87__prepower,k5_numbers,k1_numbers) ), inference(consider,[status(thm),assumptions([])],[dh_c6_87__prepower,e8_87__prepower]), [interesting(0.8),file(prepower,c6_87__prepower),[file(prepower,c6_87__prepower)]]). fof(dh_c1_87_4__prepower,definition, ( ( m2_subset_1(c1_87_4__prepower,k1_numbers,k5_numbers) => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k10_seq_1(c5_87__prepower,c6_87__prepower),c1_87_4__prepower),k10_prepower(c3_87__prepower,c1_87_4__prepower)) & r1_xreal_0(k10_prepower(c3_87__prepower,c1_87_4__prepower),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k10_seq_1(c5_87__prepower,c6_87__prepower),A),k10_prepower(c3_87__prepower,A)) & r1_xreal_0(k10_prepower(c3_87__prepower,A),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,A)) ) ) ), introduced(definition,[new_symbol(c1_87_4__prepower),file(prepower,c1_87_4__prepower)]), [interesting(0.65),axiom,file(prepower,c1_87_4__prepower)]). fof(dt_c1_87_4__prepower,assumption,( m2_subset_1(c1_87_4__prepower,k1_numbers,k5_numbers) ), introduced(assumption,[file(prepower,c1_87_4__prepower)]), [interesting(0.65),axiom,file(prepower,c1_87_4__prepower)]). fof(fc14_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc14_xreal_0)]). fof(fc15_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc15_xreal_0)]). fof(fc16_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc16_xreal_0)]). fof(fc19_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc19_xreal_0)]). fof(fc20_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc20_xreal_0)]). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(fc11_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), file(rat_1,fc11_rat_1), [interesting(0.9),axiom,file(rat_1,fc11_rat_1)]). fof(fc13_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc13_xreal_0)]). fof(fc14_rat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_rat_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), file(rat_1,fc14_rat_1), [interesting(0.9),axiom,file(rat_1,fc14_rat_1)]). fof(fc16_rat_1,theorem,( ! [A] : ( v1_rat_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_rat_1(k4_xcmplx_0(A)) ) ) ), file(rat_1,fc16_rat_1), [interesting(0.9),axiom,file(rat_1,fc16_rat_1)]). fof(fc17_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc17_xreal_0)]). fof(fc18_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc18_xreal_0)]). fof(fc1_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc1_xreal_0)]). fof(fc3_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), [interesting(0.9),axiom,file(int_1,fc3_int_1)]). fof(fc3_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), file(rat_1,fc3_rat_1), [interesting(0.9),axiom,file(rat_1,fc3_rat_1)]). fof(fc4_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), [interesting(0.9),axiom,file(int_1,fc4_int_1)]). fof(fc5_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), [interesting(0.9),axiom,file(int_1,fc5_int_1)]). fof(fc5_rat_1,theorem,( ! [A,B] : ( ( v1_rat_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), file(rat_1,fc5_rat_1), [interesting(0.9),axiom,file(rat_1,fc5_rat_1)]). fof(fc5_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc5_xreal_0)]). fof(fc8_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), [interesting(0.9),axiom,file(int_1,fc8_int_1)]). fof(fc8_rat_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_rat_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_rat_1(k6_xcmplx_0(A,B)) ) ) ), file(rat_1,fc8_rat_1), [interesting(0.9),axiom,file(rat_1,fc8_rat_1)]). fof(fc9_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), [interesting(0.9),axiom,file(int_1,fc9_int_1)]). fof(spc1_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,k4_xcmplx_0(B)) = k6_xcmplx_0(A,B) ) ), file(arithm,spc1_arithm), [interesting(0.9),axiom,file(arithm,spc1_arithm)]). fof(spc2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,k4_xcmplx_0(1)) = k4_xcmplx_0(A) ) ), file(arithm,spc2_arithm), [interesting(0.9),axiom,file(arithm,spc2_arithm)]). fof(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(redefinition_k5_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) ), file(real_1,k5_real_1), [interesting(0.9),axiom,file(real_1,k5_real_1)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k5_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k5_real_1(A,B),k1_numbers) ) ), file(real_1,k5_real_1), [interesting(0.9),axiom,file(real_1,k5_real_1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm1,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1)]). fof(rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,theorem,( k7_xcmplx_0(1,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1)]). fof(rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,theorem,( k7_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r0_r0,theorem,( k2_xcmplx_0(0,0) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm1_r0,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r1_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),1) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0)]). fof(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(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(rqLessOrEqual__r1_xreal_0__r0_r1,theorem,( r1_xreal_0(0,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm1,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r0,theorem,( ~ r1_xreal_0(1,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,( k6_xcmplx_0(0,0) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,( k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,( k6_xcmplx_0(1,0) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(rqRealDiv__k7_xcmplx_0__r0_r1_r0,theorem,( k7_xcmplx_0(0,1) = 0 ), file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r0_r0,theorem,( k3_xcmplx_0(0,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r0_r0)]). fof(rqRealMult__k3_xcmplx_0__r0_r1_r0,theorem,( k3_xcmplx_0(0,1) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r0_r1_r0)]). fof(rqRealMult__k3_xcmplx_0__r1_r0_r0,theorem,( k3_xcmplx_0(1,0) = 0 ), file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealMult__k3_xcmplx_0__r1_r0_r0)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(t2_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k3_xcmplx_0(A,0) = 0 ) ), file(arithm,t2_arithm), [interesting(0.9),axiom,file(arithm,t2_arithm)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(t5_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), [interesting(0.9),axiom,file(arithm,t5_arithm)]). 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(fc26_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & ~ v3_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc26_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc26_xreal_0)]). fof(fc18_rat_1,theorem,( ! [A] : ( v1_rat_1(A) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & v1_rat_1(k5_xcmplx_0(A)) ) ) ), file(rat_1,fc18_rat_1), [interesting(0.9),axiom,file(rat_1,fc18_rat_1)]). fof(fc25_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) & ~ v2_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc25_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc25_xreal_0)]). fof(fc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k5_xcmplx_0(A)) & v1_xreal_0(k5_xcmplx_0(A)) ) ) ), file(xreal_0,fc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc2_xreal_0)]). fof(involutiveness_k2_real_1,theorem,( ! [A] : ( m1_subset_1(A,k1_numbers) => k2_real_1(k2_real_1(A)) = A ) ), file(real_1,k2_real_1), [interesting(0.9),axiom,file(real_1,k2_real_1)]). fof(involutiveness_k5_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k5_xcmplx_0(k5_xcmplx_0(A)) = A ) ), file(xcmplx_0,k5_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k5_xcmplx_0)]). fof(redefinition_k2_real_1,definition,( ! [A] : ( m1_subset_1(A,k1_numbers) => k2_real_1(A) = k5_xcmplx_0(A) ) ), file(real_1,k2_real_1), [interesting(0.9),axiom,file(real_1,k2_real_1)]). fof(dt_k2_real_1,axiom,( ! [A] : ( m1_subset_1(A,k1_numbers) => m1_subset_1(k2_real_1(A),k1_numbers) ) ), file(real_1,k2_real_1), [interesting(0.9),axiom,file(real_1,k2_real_1)]). fof(dt_k5_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k5_xcmplx_0(A)) ) ), file(xcmplx_0,k5_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k5_xcmplx_0)]). fof(spc10_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(k5_xcmplx_0(A),k5_xcmplx_0(B)) = k5_xcmplx_0(k3_xcmplx_0(A,B)) ) ), file(arithm,spc10_arithm), [interesting(0.9),axiom,file(arithm,spc10_arithm)]). fof(spc11_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k7_xcmplx_0(k5_xcmplx_0(A),k5_xcmplx_0(B)) = k7_xcmplx_0(B,A) ) ), file(arithm,spc11_arithm), [interesting(0.9),axiom,file(arithm,spc11_arithm)]). fof(spc12_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k3_xcmplx_0(A,k5_xcmplx_0(B)) = k7_xcmplx_0(A,B) ) ), file(arithm,spc12_arithm), [interesting(0.9),axiom,file(arithm,spc12_arithm)]). fof(spc3_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(1,A) = k5_xcmplx_0(A) ) ), file(arithm,spc3_arithm), [interesting(0.9),axiom,file(arithm,spc3_arithm)]). fof(t52_int_1,theorem,( ! [A] : ( v1_xreal_0(A) => ~ r1_xreal_0(k2_xcmplx_0(k1_int_1(A),1),A) ) ), file(int_1,t52_int_1), [interesting(0.9),axiom,file(int_1,t52_int_1)]). fof(e7_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),k2_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc1_int_1,fc23_xreal_0,fc2_int_1,fc2_membered,fc6_int_1,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,cc2_xreal_0,fc3_xreal_0,fc4_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_numerals,spc1_boole,t52_int_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e7_87_4__prepower),[file(prepower,e7_87_4__prepower)]]). fof(t11_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), [interesting(0.9),axiom,file(xreal_1,t11_xreal_1)]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(e8_87_4__prepower,plain,( r1_xreal_0(k6_xcmplx_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),1),k6_xcmplx_0(k2_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),1),1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_rat_1,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc14_rat_1,fc14_xreal_0,fc15_rat_1,fc15_xreal_0,fc16_rat_1,fc16_xreal_0,fc19_xreal_0,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc3_rat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_int_1,fc23_xreal_0,fc2_int_1,fc2_membered,fc3_int_1,fc4_int_1,fc5_int_1,fc6_int_1,fc7_int_1,fc8_int_1,fc8_xreal_0,fc9_int_1,rc1_int_1,rc1_xreal_0,rc2_int_1,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e7_87_4__prepower,t11_xreal_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(prepower,e8_87_4__prepower),[file(prepower,e8_87_4__prepower)]]). fof(t66_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(0,C) ) => r1_xreal_0(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t66_xreal_1), [interesting(0.9),axiom,file(xreal_1,t66_xreal_1)]). fof(e9_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),1),k2_real_1(k1_nat_1(c1_87_4__prepower,1))),k3_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k2_real_1(k1_nat_1(c1_87_4__prepower,1)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_rat_1,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc14_rat_1,fc14_xreal_0,fc15_rat_1,fc15_xreal_0,fc16_rat_1,fc16_xreal_0,fc18_rat_1,fc19_xreal_0,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc3_rat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,involutiveness_k5_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_k5_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_int_1,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_membered,fc2_xreal_0,fc3_int_1,fc4_int_1,fc5_int_1,fc6_int_1,fc7_int_1,fc8_int_1,fc8_xreal_0,fc9_int_1,rc1_int_1,rc1_xreal_0,rc2_int_1,spc10_arithm,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_real_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_real_1,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e8_87_4__prepower,t66_xreal_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(prepower,e9_87_4__prepower),[file(prepower,e9_87_4__prepower)]]). fof(d9_xcmplx_0,definition,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => k7_xcmplx_0(A,B) = k3_xcmplx_0(A,k5_xcmplx_0(B)) ) ) ), file(xcmplx_0,d9_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,d9_xcmplx_0)]). fof(e10_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),1),k2_real_1(k1_nat_1(c1_87_4__prepower,1))),k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k1_nat_1(c1_87_4__prepower,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_rat_1,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc14_rat_1,fc14_xreal_0,fc15_rat_1,fc15_xreal_0,fc16_rat_1,fc16_xreal_0,fc17_rat_1,fc18_rat_1,fc19_xreal_0,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc3_rat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_int_1,fc1_xreal_0,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_membered,fc2_xreal_0,fc30_xreal_0,fc3_int_1,fc3_xreal_0,fc4_int_1,fc4_xreal_0,fc5_int_1,fc5_xreal_0,fc6_int_1,fc6_xreal_0,fc7_int_1,fc8_int_1,fc8_xreal_0,fc9_int_1,rc1_int_1,rc1_xreal_0,rc2_int_1,spc1_arithm,spc5_arithm,spc6_arithm,spc8_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,involutiveness_k5_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_real_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_xcmplx_0,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc10_arithm,spc11_arithm,spc12_arithm,spc2_arithm,spc3_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t3_arithm,t6_arithm,spc1_numerals,spc1_boole,e9_87_4__prepower,d9_xcmplx_0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.65),file(prepower,e10_87_4__prepower),[file(prepower,e10_87_4__prepower)]]). fof(e11_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),1),k2_real_1(k1_nat_1(c1_87_4__prepower,1))),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_nat_1,fc1_seq_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc3_nat_1,fc4_nat_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_prepower,rc1_seq_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,involutiveness_k5_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_k5_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_rat_1,fc18_xreal_0,fc1_int_1,fc1_ordinal2,fc1_rat_1,fc1_xreal_0,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_rat_1,fc2_xreal_0,fc30_xreal_0,fc3_int_1,fc3_rat_1,fc3_xreal_0,fc4_int_1,fc4_rat_1,fc4_xreal_0,fc5_int_1,fc5_membered,fc5_rat_1,fc5_xreal_0,fc6_int_1,fc6_rat_1,fc6_xreal_0,fc7_int_1,fc7_rat_1,fc8_int_1,fc8_rat_1,fc8_xreal_0,fc9_int_1,fc9_rat_1,rc1_int_1,rc1_rat_1,rc1_xreal_0,rc2_int_1,rc2_rat_1,spc10_arithm,spc11_arithm,spc12_arithm,spc1_arithm,spc2_arithm,spc3_arithm,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k2_real_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_prepower,dt_k1_int_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_real_1,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c2_87__prepower,dt_c3_87__prepower,de_c3_87__prepower,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,spc1_boole,e10_87_4__prepower,e2_87__prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.65),file(prepower,e11_87_4__prepower),[file(prepower,e11_87_4__prepower)]]). fof(e12_87_4__prepower,plain,( r1_xreal_0(k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),1),k1_nat_1(c1_87_4__prepower,1)),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,fc1_seq_1,rc1_prepower,rc1_seq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_int_1,fc6_membered,fc6_rat_1,fc7_int_1,fc7_rat_1,fc7_xreal_0,fc8_int_1,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c2_87__prepower,cc15_membered,cc1_nat_1,cc1_rat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_rat_1,fc18_xreal_0,fc1_rat_1,fc1_xreal_0,fc23_xreal_0,fc25_xreal_0,fc2_membered,fc2_rat_1,fc2_xreal_0,fc30_xreal_0,fc3_rat_1,fc3_xreal_0,fc4_xreal_0,fc5_int_1,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,spc1_arithm,spc5_arithm,spc6_arithm,spc8_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,involutiveness_k5_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k2_real_1,dt_k10_prepower,dt_k1_nat_1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_xcmplx_0,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c3_87__prepower,de_c3_87__prepower,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc10_arithm,spc11_arithm,spc12_arithm,spc2_arithm,spc3_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t3_arithm,t6_arithm,spc1_numerals,spc1_boole,e11_87_4__prepower,d9_xcmplx_0,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.65),file(prepower,e12_87_4__prepower),[file(prepower,e12_87_4__prepower)]]). fof(t121_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k6_xcmplx_0(k7_xcmplx_0(A,B),k7_xcmplx_0(C,B)) = k7_xcmplx_0(k6_xcmplx_0(A,C),B) ) ) ) ), file(xcmplx_1,t121_xcmplx_1), [interesting(0.9),axiom,file(xcmplx_1,t121_xcmplx_1)]). fof(e13_87_4__prepower,plain,( r1_xreal_0(k6_xcmplx_0(k7_xcmplx_0(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower),k1_nat_1(c1_87_4__prepower,1)),k6_real_1(1,k1_nat_1(c1_87_4__prepower,1))),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,fc1_seq_1,rc1_prepower,rc1_seq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_int_1,fc6_membered,fc6_rat_1,fc7_int_1,fc7_rat_1,fc7_xreal_0,fc8_int_1,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c2_87__prepower,cc15_membered,cc1_nat_1,cc1_rat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_rat_1,fc1_xreal_0,fc23_xreal_0,fc2_membered,fc2_rat_1,fc30_xreal_0,fc3_rat_1,fc3_xreal_0,fc4_xreal_0,fc5_int_1,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,spc1_arithm,spc5_arithm,spc6_arithm,spc8_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k10_prepower,dt_k1_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c3_87__prepower,de_c3_87__prepower,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t3_arithm,t6_arithm,spc1_numerals,spc1_boole,e12_87_4__prepower,t121_xcmplx_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e13_87_4__prepower),[file(prepower,e13_87_4__prepower)]]). fof(t75_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => k3_xcmplx_0(A,k7_xcmplx_0(B,C)) = k7_xcmplx_0(k3_xcmplx_0(A,B),C) ) ) ) ), file(xcmplx_1,t75_xcmplx_1), [interesting(0.9),axiom,file(xcmplx_1,t75_xcmplx_1)]). fof(e14_87_4__prepower,plain,( r1_xreal_0(k6_xcmplx_0(k3_xcmplx_0(k7_xcmplx_0(c1_87__prepower,k1_nat_1(c1_87_4__prepower,1)),k1_nat_1(c1_87_4__prepower,1)),k6_real_1(1,k1_nat_1(c1_87_4__prepower,1))),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,fc1_seq_1,rc1_prepower,rc1_seq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_int_1,fc6_membered,fc6_rat_1,fc7_int_1,fc7_rat_1,fc7_xreal_0,fc8_int_1,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c2_87__prepower,cc15_membered,cc1_nat_1,cc1_rat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_rat_1,fc1_xreal_0,fc23_xreal_0,fc2_membered,fc2_rat_1,fc30_xreal_0,fc3_rat_1,fc3_xreal_0,fc4_xreal_0,fc5_int_1,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,spc1_arithm,spc5_arithm,spc6_arithm,spc8_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k10_prepower,dt_k1_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c3_87__prepower,de_c3_87__prepower,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t3_arithm,t6_arithm,spc1_numerals,spc1_boole,e13_87_4__prepower,t75_xcmplx_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e14_87_4__prepower),[file(prepower,e14_87_4__prepower)]]). fof(t88_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( A != 0 => k3_xcmplx_0(k7_xcmplx_0(B,A),A) = B ) ) ) ), file(xcmplx_1,t88_xcmplx_1), [interesting(0.9),axiom,file(xcmplx_1,t88_xcmplx_1)]). fof(e15_87_4__prepower,plain,( r1_xreal_0(k6_xcmplx_0(c1_87__prepower,k6_real_1(1,k1_nat_1(c1_87_4__prepower,1))),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[reflexivity_r1_tarski,fc1_seq_1,rc1_prepower,rc1_seq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc5_rat_1,fc6_int_1,fc6_membered,fc6_rat_1,fc7_int_1,fc7_rat_1,fc7_xreal_0,fc8_int_1,fc8_rat_1,fc9_rat_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c2_87__prepower,cc15_membered,cc1_nat_1,cc1_rat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_rat_1,fc1_xreal_0,fc23_xreal_0,fc2_membered,fc2_rat_1,fc30_xreal_0,fc3_rat_1,fc3_xreal_0,fc4_xreal_0,fc5_int_1,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,spc1_arithm,spc5_arithm,spc6_arithm,spc8_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k6_real_1,dt_k10_prepower,dt_k1_nat_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k6_real_1,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c3_87__prepower,de_c3_87__prepower,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc2_arithm,spc4_arithm,spc7_arithm,spc9_arithm,t2_arithm,t3_arithm,t4_arithm,t5_arithm,t6_arithm,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e14_87_4__prepower,t88_xcmplx_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e15_87_4__prepower),[file(prepower,e15_87_4__prepower)]]). fof(t13_funcop_1,theorem,( ! [A,B,C] : ( r2_hidden(B,A) => k1_funct_1(k2_funcop_1(A,C),B) = C ) ), file(funcop_1,t13_funcop_1), [interesting(0.9),axiom,file(funcop_1,t13_funcop_1)]). fof(e7_87__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,A) = c4_87__prepower ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_rat_1,rc2_rat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_seqm_3,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_seq_1,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc3_seqm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k2_funcop_1,dt_k2_seq_1,dt_k5_numbers,dt_m2_subset_1,dt_c4_87__prepower,dt_c5_87__prepower,de_c4_87__prepower,de_c5_87__prepower,fc2_membered,fc2_seqm_3,t1_subset,t7_boole,t13_funcop_1]), [interesting(0.8),file(prepower,e7_87__prepower),[file(prepower,e7_87__prepower)]]). fof(e16_87_4__prepower,plain,( r1_xreal_0(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower),k6_real_1(1,k1_nat_1(c1_87_4__prepower,1))),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc4_ordinal2,fc4_rat_1,fc5_rat_1,fc6_int_1,fc6_membered,fc7_rat_1,fc7_xreal_0,fc8_int_1,fc8_rat_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_seq_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_k7_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c2_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc10_rat_1,fc11_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_xreal_0,fc2_rat_1,fc2_seqm_3,fc30_xreal_0,fc3_rat_1,fc3_xreal_0,fc5_int_1,fc5_membered,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_real,t2_subset,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k10_prepower,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_k6_real_1,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c3_87__prepower,dt_c4_87__prepower,dt_c5_87__prepower,de_c3_87__prepower,de_c4_87__prepower,de_c5_87__prepower,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,spc1_boole,e15_87_4__prepower,e7_87__prepower,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.65),file(prepower,e16_87_4__prepower),[file(prepower,e16_87_4__prepower)]]). fof(e9_87__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c6_87__prepower,A) = k6_real_1(1,k1_nat_1(A,1)) ) ), inference(consider,[status(thm),assumptions([])],[dh_c6_87__prepower,e8_87__prepower]), [interesting(0.8),file(prepower,e9_87__prepower),[file(prepower,e9_87__prepower)]]). fof(e17_87_4__prepower,plain,( r1_xreal_0(k5_real_1(k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower),k2_seq_1(k5_numbers,k1_numbers,c6_87__prepower,c1_87_4__prepower)),k10_prepower(c3_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_c1_87__prepower,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc4_ordinal2,fc4_rat_1,fc5_rat_1,fc6_int_1,fc6_membered,fc7_rat_1,fc7_xreal_0,fc8_int_1,fc8_rat_1,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_seq_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c2_87__prepower,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc10_rat_1,fc11_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_xreal_0,fc2_rat_1,fc2_seqm_3,fc30_xreal_0,fc3_rat_1,fc3_xreal_0,fc5_int_1,fc5_membered,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_real,t2_subset,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k6_real_1,redefinition_m2_subset_1,dt_k10_prepower,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_k6_real_1,dt_m2_subset_1,dt_c1_87_4__prepower,dt_c3_87__prepower,dt_c5_87__prepower,dt_c6_87__prepower,de_c3_87__prepower,de_c5_87__prepower,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,spc1_boole,e16_87_4__prepower,e9_87__prepower,rqRealNeg__k4_xcmplx_0__r1_rm1]), [interesting(0.65),file(prepower,e17_87_4__prepower),[file(prepower,e17_87_4__prepower)]]). fof(commutativity_k4_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k4_real_1(B,A) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(redefinition_k4_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(dt_k4_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_real_1(A,B),k1_numbers) ) ), file(real_1,k4_real_1), [interesting(0.9),axiom,file(real_1,k4_real_1)]). fof(d4_int_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_int_1(B) => ( B = k1_int_1(A) <=> ( r1_xreal_0(B,A) & ~ r1_xreal_0(B,k6_xcmplx_0(A,1)) ) ) ) ) ), file(int_1,d4_int_1), [interesting(0.9),axiom,file(int_1,d4_int_1)]). fof(e1_87_4__prepower,plain,( r1_xreal_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_nat_1,fc1_ordinal2,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_nat_1,fc3_nat_1,fc4_nat_1,fc5_membered,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc1_nat_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc1_rat_1,cc2_int_1,cc2_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc14_rat_1,fc15_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_int_1,fc1_rat_1,fc23_xreal_0,fc2_membered,fc2_rat_1,fc3_rat_1,fc3_xreal_0,fc4_rat_1,fc5_rat_1,fc6_int_1,fc6_rat_1,fc7_int_1,fc7_rat_1,fc8_int_1,fc8_rat_1,fc8_xreal_0,fc9_int_1,fc9_rat_1,rc1_int_1,rc1_rat_1,rc1_xreal_0,rc2_rat_1,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_int_1,dt_k1_nat_1,dt_k3_xcmplx_0,dt_k6_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,cc2_rat_1,cc2_xreal_0,cc4_int_1,fc2_int_1,fc4_int_1,fc4_xreal_0,fc5_xreal_0,rc2_int_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_numerals,spc1_boole,d4_int_1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e1_87_4__prepower),[file(prepower,e1_87_4__prepower)]]). fof(e2_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k2_real_1(k1_nat_1(c1_87_4__prepower,1))),k3_xcmplx_0(k3_xcmplx_0(c1_87__prepower,k1_nat_1(c1_87_4__prepower,1)),k2_real_1(k1_nat_1(c1_87_4__prepower,1)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc18_rat_1,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,involutiveness_k5_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_k5_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc1_int_1,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_membered,fc2_xreal_0,fc3_xreal_0,fc6_int_1,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc10_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_arithm,t1_numerals,t1_real,t2_arithm,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_real_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_real_1,dt_k3_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,cc2_xreal_0,fc4_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_87_4__prepower,t66_xreal_1,rqRealMult__k3_xcmplx_0__r1_r1_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(prepower,e2_87_4__prepower),[file(prepower,e2_87_4__prepower)]]). fof(e3_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k2_real_1(k1_nat_1(c1_87_4__prepower,1))),k3_xcmplx_0(c1_87__prepower,k4_real_1(k1_nat_1(c1_87_4__prepower,1),k2_real_1(k1_nat_1(c1_87_4__prepower,1))))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc18_rat_1,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,involutiveness_k5_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_k5_xcmplx_0,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc1_int_1,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_membered,fc2_xreal_0,fc3_xreal_0,fc4_xreal_0,fc6_int_1,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,spc10_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_real_1,redefinition_k4_real_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_c1_87__prepower,dt_c1_87_4__prepower,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_numerals,spc1_boole,e2_87_4__prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e3_87_4__prepower),[file(prepower,e3_87_4__prepower)]]). fof(d7_xcmplx_0,definition,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ( ( A != 0 => ( B = k5_xcmplx_0(A) <=> k3_xcmplx_0(A,B) = 1 ) ) & ( A = 0 => ( B = k5_xcmplx_0(A) <=> B = 0 ) ) ) ) ) ), file(xcmplx_0,d7_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,d7_xcmplx_0)]). fof(e4_87_4__prepower,plain,( r1_xreal_0(k3_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k2_real_1(k1_nat_1(c1_87_4__prepower,1))),k3_xcmplx_0(c1_87__prepower,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc18_rat_1,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc1_int_1,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_membered,fc2_xreal_0,fc3_xreal_0,fc4_xreal_0,fc6_int_1,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc5_arithm,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,commutativity_k4_real_1,involutiveness_k5_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_real_1,redefinition_k4_real_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k4_real_1,dt_k5_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,spc10_arithm,spc7_arithm,t2_arithm,t3_arithm,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e3_87_4__prepower,d7_xcmplx_0,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e4_87_4__prepower),[file(prepower,e4_87_4__prepower)]]). fof(e5_87_4__prepower,plain,( r1_xreal_0(k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k1_nat_1(c1_87_4__prepower,1)),c1_87__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower,dt_c1_87_4__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc17_rat_1,fc18_rat_1,fc1_nat_1,fc1_ordinal2,fc1_rat_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc26_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc6_xreal_0,cc7_xreal_0,fc1_int_1,fc23_xreal_0,fc25_xreal_0,fc2_int_1,fc2_membered,fc2_xreal_0,fc30_xreal_0,fc3_xreal_0,fc4_xreal_0,fc6_int_1,fc6_xreal_0,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,spc5_arithm,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k2_real_1,commutativity_k3_xcmplx_0,involutiveness_k5_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k2_real_1,dt_k1_int_1,dt_k1_nat_1,dt_k2_real_1,dt_k3_xcmplx_0,dt_k5_xcmplx_0,dt_k7_xcmplx_0,dt_c1_87__prepower,dt_c1_87_4__prepower,rqLessOrEqual__r1_xreal_0__r1_r1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc10_arithm,spc11_arithm,spc12_arithm,spc3_arithm,spc4_arithm,spc7_arithm,t3_arithm,t6_arithm,spc1_numerals,spc1_boole,e4_87_4__prepower,d9_xcmplx_0,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e5_87_4__prepower),[file(prepower,e5_87_4__prepower)]]). fof(e6_87_4__prepower,plain,( r1_xreal_0(k7_xcmplx_0(k1_int_1(k3_xcmplx_0(k1_nat_1(c1_87_4__prepower,1),c1_87__prepower)),k1_nat_1(c1_87_4__prepower,1)),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_rat_1,fc10_xreal_0,fc11_xreal_0,fc12_rat_1,fc12_xreal_0,fc13_rat_1,fc15_rat_1,fc17_rat_1,fc1_nat_1,fc1_rat_1,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc2_rat_1,fc3_nat_1,fc4_nat_1,fc4_ordinal2,fc4_rat_1,fc6_membered,fc6_rat_1,fc7_rat_1,fc7_xreal_0,fc9_rat_1,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_rat_1,rc1_seq_1,rc2_nat_1,rc2_rat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_int_1,fc1_ordinal2,fc23_xreal_0,fc2_int_1,fc2_seqm_3,fc30_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc6_int_1,fc6_xreal_0,fc7_int_1,fc8_xreal_0,rc1_int_1,rc1_xreal_0,rc2_int_1,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_int_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c4_87__prepower,dt_c5_87__prepower,de_c4_87__prepower,de_c5_87__prepower,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,spc1_numerals,spc1_boole,e5_87_4__prepower,e7_87__prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e6_87_4__prepower),[file(prepower,e6_87_4__prepower)]]). fof(t6_rfunct_2,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( A = k10_seq_1(B,C) <=> ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,D) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,B,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) ) ) ) ) ) ), file(rfunct_2,t6_rfunct_2), [interesting(0.9),axiom,file(rfunct_2,t6_rfunct_2)]). fof(e18_87_4__prepower,plain, ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k10_seq_1(c5_87__prepower,c6_87__prepower),c1_87_4__prepower),k10_prepower(c3_87__prepower,c1_87_4__prepower)) & r1_xreal_0(k10_prepower(c3_87__prepower,c1_87_4__prepower),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_nat_1,fc1_seq_1,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_nat_1,fc3_nat_1,fc4_nat_1,fc4_ordinal2,fc4_seq_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_membered,rc1_nat_1,rc1_prepower,rc1_seq_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t2_real,t3_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k4_seq_1,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_rat_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc13_xreal_0,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc17_rat_1,fc17_xreal_0,fc18_xreal_0,fc1_int_1,fc1_ordinal2,fc1_rat_1,fc1_xreal_0,fc23_xreal_0,fc2_int_1,fc2_rat_1,fc2_seqm_3,fc30_xreal_0,fc3_int_1,fc3_rat_1,fc3_xreal_0,fc4_int_1,fc4_rat_1,fc4_xreal_0,fc5_int_1,fc5_membered,fc5_rat_1,fc5_xreal_0,fc6_int_1,fc6_rat_1,fc6_xreal_0,fc7_int_1,fc7_rat_1,fc8_int_1,fc8_rat_1,fc8_xreal_0,fc9_int_1,fc9_rat_1,rc1_int_1,rc1_rat_1,rc1_xreal_0,rc2_int_1,rc2_rat_1,spc1_arithm,spc2_arithm,spc4_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_real,t2_subset,t3_arithm,t3_subset,t4_real,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k3_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_relset_1,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k10_seq_1,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_prepower,dt_k10_seq_1,dt_k1_int_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_k7_xcmplx_0,dt_m2_relset_1,dt_m2_subset_1,dt_c1_87__prepower,dt_c1_87_4__prepower,dt_c2_87__prepower,dt_c3_87__prepower,dt_c5_87__prepower,dt_c6_87__prepower,de_c3_87__prepower,de_c5_87__prepower,fc2_membered,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,spc1_numerals,spc1_boole,e17_87_4__prepower,e2_87__prepower,e6_87_4__prepower,t6_rfunct_2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.65),file(prepower,e18_87_4__prepower),[file(prepower,e18_87_4__prepower)]]). fof(i2_87_4__prepower,theorem,( $true ), introduced(tautology,[file(prepower,i2_87_4__prepower)]), [interesting(0.65),trivial,file(prepower,i2_87_4__prepower)]). fof(i1_87_4__prepower,plain, ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k10_seq_1(c5_87__prepower,c6_87__prepower),c1_87_4__prepower),k10_prepower(c3_87__prepower,c1_87_4__prepower)) & r1_xreal_0(k10_prepower(c3_87__prepower,c1_87_4__prepower),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower)) ), inference(conclusion,[status(thm),assumptions([dt_c1_87_4__prepower,dt_c1_87__prepower])],[dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_seq_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_rat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k4_seq_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc2_rat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_prepower,redefinition_k10_seq_1,redefinition_k2_seq_1,redefinition_k5_numbers,dt_k10_prepower,dt_k10_seq_1,dt_k1_numbers,dt_k2_seq_1,dt_k5_numbers,dt_c1_87_4__prepower,dt_c3_87__prepower,dt_c5_87__prepower,dt_c6_87__prepower,fc2_membered,e18_87_4__prepower,i2_87_4__prepower]), [interesting(0.65),file(prepower,i1_87_4__prepower),[file(prepower,i1_87_4__prepower)]]). fof(i1_87_4_tmp__prepower,plain, ( m2_subset_1(c1_87_4__prepower,k1_numbers,k5_numbers) => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k10_seq_1(c5_87__prepower,c6_87__prepower),c1_87_4__prepower),k10_prepower(c3_87__prepower,c1_87_4__prepower)) & r1_xreal_0(k10_prepower(c3_87__prepower,c1_87_4__prepower),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c1_87_4__prepower)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_87__prepower]),discharge_asm(discharge,[dt_c1_87_4__prepower])],[dt_c1_87_4__prepower,i1_87_4__prepower]), [interesting(0.8),e15_87__prepower]). fof(e15_87__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,k10_seq_1(c5_87__prepower,c6_87__prepower),A),k10_prepower(c3_87__prepower,A)) & r1_xreal_0(k10_prepower(c3_87__prepower,A),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,A)) ) ) ), inference(let,[status(thm),assumptions([dt_c1_87__prepower])],[i1_87_4_tmp__prepower,dh_c1_87_4__prepower]), [interesting(0.8),file(prepower,e15_87__prepower),[file(prepower,e15_87__prepower)]]). fof(t39_seq_4,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v5_seqm_3(A) => v4_seq_2(A) ) ) ), file(seq_4,t39_seq_4), [interesting(0.9),axiom,file(seq_4,t39_seq_4)]). fof(e11_87__prepower,plain,( v4_seq_2(c5_87__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_rat_1,rc2_rat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c1_87__prepower,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_seqm_3,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc3_seqm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_relset_1,dt_c5_87__prepower,de_c5_87__prepower,fc2_membered,t39_seq_4]), [interesting(0.8),file(prepower,e11_87__prepower),[file(prepower,e11_87__prepower)]]). fof(redefinition_k8_funct_2,definition,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), [interesting(0.9),axiom,file(funct_2,k8_funct_2)]). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), [interesting(0.9),axiom,file(funct_2,k8_funct_2)]). fof(t45_seq_4,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k8_funct_2(k5_numbers,k1_numbers,A,B) = k6_real_1(1,k1_nat_1(B,1)) ) => ( v4_seq_2(A) & k2_seq_2(A) = 0 ) ) ) ), file(seq_4,t45_seq_4), [interesting(0.9),axiom,file(seq_4,t45_seq_4)]). fof(e10_87__prepower,plain, ( v4_seq_2(c6_87__prepower) & k2_seq_2(c6_87__prepower) = 0 ), inference(mizar_by,[status(thm),assumptions([])],[cc1_rat_1,fc10_rat_1,fc13_rat_1,fc17_rat_1,fc2_rat_1,fc4_rat_1,fc7_rat_1,rc1_rat_1,rc2_rat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,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,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc30_xreal_0,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc6_membered,fc6_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_arithm,t1_subset,t4_subset,t5_arithm,t5_subset,t6_arithm,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_k7_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc5_membered,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k5_numbers,redefinition_k6_real_1,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_seq_2,dt_k5_numbers,dt_k6_real_1,dt_k8_funct_2,dt_m2_relset_1,dt_m2_subset_1,dt_c6_87__prepower,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e9_87__prepower,t45_seq_4]), [interesting(0.8),file(prepower,e10_87__prepower),[file(prepower,e10_87__prepower)]]). fof(t25_seq_2,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( ( v4_seq_2(A) & v4_seq_2(B) ) => v4_seq_2(k10_seq_1(A,B)) ) ) ) ), file(seq_2,t25_seq_2), [interesting(0.9),axiom,file(seq_2,t25_seq_2)]). fof(e12_87__prepower,plain,( v4_seq_2(k10_seq_1(c5_87__prepower,c6_87__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc4_seqm_3,rc1_rat_1,rc2_rat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c1_87__prepower,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc4_ordinal2,fc4_seq_1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_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_relset_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_m2_subset_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_zfmisc_1,dt_k4_seq_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_subset_1,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,redefinition_k10_seq_1,redefinition_k2_seq_2,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k10_seq_1,dt_k1_numbers,dt_k2_seq_2,dt_k5_numbers,dt_m2_relset_1,dt_c5_87__prepower,dt_c6_87__prepower,de_c5_87__prepower,fc2_membered,spc0_numerals,spc0_boole,e11_87__prepower,e10_87__prepower,t25_seq_2]), [interesting(0.8),file(prepower,e12_87__prepower),[file(prepower,e12_87__prepower)]]). fof(fc3_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k3_seq_1(A,B)) & v1_funct_1(k3_seq_1(A,B)) & v1_seq_1(k3_seq_1(A,B)) ) ) ), file(seq_1,fc3_seq_1), [interesting(0.9),axiom,file(seq_1,fc3_seq_1)]). fof(fc5_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k5_seq_1(A,B)) & v1_funct_1(k5_seq_1(A,B)) & v1_seq_1(k5_seq_1(A,B)) ) ) ), file(seq_1,fc5_seq_1), [interesting(0.9),axiom,file(seq_1,fc5_seq_1)]). fof(commutativity_k3_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => k3_seq_1(A,B) = k3_seq_1(B,A) ) ), file(seq_1,k3_seq_1), [interesting(0.9),axiom,file(seq_1,k3_seq_1)]). fof(commutativity_k5_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => k5_seq_1(A,B) = k5_seq_1(B,A) ) ), file(seq_1,k5_seq_1), [interesting(0.9),axiom,file(seq_1,k5_seq_1)]). fof(dt_k3_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k3_seq_1(A,B)) & v1_funct_1(k3_seq_1(A,B)) ) ) ), file(seq_1,k3_seq_1), [interesting(0.9),axiom,file(seq_1,k3_seq_1)]). fof(dt_k5_seq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_seq_1(B) ) => ( v1_relat_1(k5_seq_1(A,B)) & v1_funct_1(k5_seq_1(A,B)) ) ) ), file(seq_1,k5_seq_1), [interesting(0.9),axiom,file(seq_1,k5_seq_1)]). fof(commutativity_k11_seq_1,theorem,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k11_seq_1(A,B) = k11_seq_1(B,A) ) ), file(seq_1,k11_seq_1), [interesting(0.9),axiom,file(seq_1,k11_seq_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(commutativity_k9_seq_1,theorem,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k9_seq_1(A,B) = k9_seq_1(B,A) ) ), file(seq_1,k9_seq_1), [interesting(0.9),axiom,file(seq_1,k9_seq_1)]). fof(redefinition_k11_seq_1,definition,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k11_seq_1(A,B) = k5_seq_1(A,B) ) ), file(seq_1,k11_seq_1), [interesting(0.9),axiom,file(seq_1,k11_seq_1)]). fof(redefinition_k3_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(redefinition_k9_seq_1,definition,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => k9_seq_1(A,B) = k3_seq_1(A,B) ) ), file(seq_1,k9_seq_1), [interesting(0.9),axiom,file(seq_1,k9_seq_1)]). fof(dt_k11_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_funct_1(k11_seq_1(A,B)) & v1_funct_2(k11_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k11_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k11_seq_1), [interesting(0.9),axiom,file(seq_1,k11_seq_1)]). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(dt_k9_seq_1,axiom,( ! [A,B] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m1_relset_1(A,k5_numbers,k1_numbers) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m1_relset_1(B,k5_numbers,k1_numbers) ) => ( v1_funct_1(k9_seq_1(A,B)) & v1_funct_2(k9_seq_1(A,B),k5_numbers,k1_numbers) & m2_relset_1(k9_seq_1(A,B),k5_numbers,k1_numbers) ) ) ), file(seq_1,k9_seq_1), [interesting(0.9),axiom,file(seq_1,k9_seq_1)]). fof(t59_seq_4,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( ( v5_seqm_3(A) & v4_seq_2(B) ) => ( k2_seq_2(k9_seq_1(A,B)) = k3_real_1(k8_funct_2(k5_numbers,k1_numbers,A,0),k2_seq_2(B)) & k2_seq_2(k10_seq_1(A,B)) = k5_real_1(k8_funct_2(k5_numbers,k1_numbers,A,0),k2_seq_2(B)) & k2_seq_2(k10_seq_1(B,A)) = k5_real_1(k2_seq_2(B),k8_funct_2(k5_numbers,k1_numbers,A,0)) & k2_seq_2(k11_seq_1(A,B)) = k4_real_1(k8_funct_2(k5_numbers,k1_numbers,A,0),k2_seq_2(B)) ) ) ) ) ), file(seq_4,t59_seq_4), [interesting(0.9),axiom,file(seq_4,t59_seq_4)]). fof(e1_87_2__prepower,plain,( k2_seq_2(k10_seq_1(c5_87__prepower,c6_87__prepower)) = k5_real_1(k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,0),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,fc10_rat_1,fc11_rat_1,fc12_rat_1,fc13_rat_1,fc14_rat_1,fc15_rat_1,fc16_rat_1,fc1_rat_1,fc2_rat_1,fc3_rat_1,fc4_rat_1,fc5_rat_1,fc6_rat_1,fc7_rat_1,fc8_rat_1,fc9_rat_1,rc1_rat_1,rc2_rat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c1_87__prepower,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_seqm_3,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc1_xreal_0,fc20_xreal_0,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_int_1,fc3_nat_1,fc3_seq_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc4_ordinal2,fc4_seq_1,fc4_xreal_0,fc5_seq_1,fc5_xreal_0,fc6_int_1,fc6_membered,fc7_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_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,commutativity_k2_xcmplx_0,commutativity_k3_seq_1,commutativity_k3_xcmplx_0,commutativity_k5_seq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_subset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_xcmplx_0,dt_k2_zfmisc_1,dt_k3_seq_1,dt_k3_xcmplx_0,dt_k4_seq_1,dt_k5_ordinal2,dt_k5_seq_1,dt_k6_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_subset_1,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc3_seqm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_int_1,fc5_membered,fc9_int_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealMult__k3_xcmplx_0__r0_r0_r0,rqRealMult__k3_xcmplx_0__r0_r1_r0,rqRealMult__k3_xcmplx_0__r1_r0_r0,rqRealMult__k3_xcmplx_0__r1_r1_r1,spc1_arithm,spc2_arithm,spc5_arithm,spc6_arithm,spc7_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_arithm,t2_subset,t3_arithm,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,commutativity_k11_seq_1,commutativity_k3_real_1,commutativity_k4_real_1,involutiveness_k4_xcmplx_0,commutativity_k9_seq_1,existence_m2_relset_1,redefinition_k10_seq_1,redefinition_k11_seq_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k3_real_1,redefinition_k4_real_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_k8_funct_2,redefinition_k9_seq_1,redefinition_m2_relset_1,dt_k10_seq_1,dt_k11_seq_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_seq_2,dt_k3_real_1,dt_k4_real_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_k8_funct_2,dt_k9_seq_1,dt_m2_relset_1,dt_c5_87__prepower,dt_c6_87__prepower,de_c5_87__prepower,fc2_membered,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e10_87__prepower,t59_seq_4,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0]), [interesting(0.65),file(prepower,e1_87_2__prepower),[file(prepower,e1_87_2__prepower)]]). fof(e2_87_2__prepower,plain,( k5_real_1(k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,0),0) = c1_87__prepower ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc4_seqm_3,fc11_rat_1,fc14_rat_1,fc16_rat_1,fc3_rat_1,fc5_rat_1,fc8_rat_1,rc1_rat_1,rc2_rat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_seq_1,fc20_xreal_0,fc3_int_1,fc4_int_1,fc4_ordinal2,fc6_membered,fc8_int_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_1,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_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_xreal_0,fc2_seqm_3,fc5_int_1,fc5_membered,fc5_xreal_0,fc9_int_1,rc1_xreal_0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,spc9_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_k5_real_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_seq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k5_real_1,dt_m2_subset_1,dt_c1_87__prepower,dt_c4_87__prepower,dt_c5_87__prepower,de_c4_87__prepower,de_c5_87__prepower,fc2_membered,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e7_87__prepower,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0]), [interesting(0.65),file(prepower,e2_87_2__prepower),[file(prepower,e2_87_2__prepower)]]). fof(e13_87__prepower,plain,( k2_seq_2(k10_seq_1(c5_87__prepower,c6_87__prepower)) = c1_87__prepower ), inference(iterative_eq,[status(thm),assumptions([dt_c1_87__prepower])],[e1_87_2__prepower,e2_87_2__prepower]), [interesting(0.8),file(prepower,e13_87__prepower),[file(prepower,e13_87__prepower)]]). fof(t41_seq_4,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( v5_seqm_3(A) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_2(A) = k8_funct_2(k5_numbers,k1_numbers,A,B) ) ) ) ), file(seq_4,t41_seq_4), [interesting(0.9),axiom,file(seq_4,t41_seq_4)]). fof(e1_87_3__prepower,plain,( k2_seq_2(c5_87__prepower) = k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,rc1_rat_1,rc2_rat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c1_87__prepower,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_seqm_3,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_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_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc3_seqm_3,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_relset_1,existence_m2_subset_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k5_numbers,redefinition_k8_funct_2,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_seq_2,dt_k5_numbers,dt_k8_funct_2,dt_m2_relset_1,dt_m2_subset_1,dt_c5_87__prepower,de_c5_87__prepower,fc2_membered,spc0_numerals,spc0_boole,t41_seq_4]), [interesting(0.65),file(prepower,e1_87_3__prepower),[file(prepower,e1_87_3__prepower)]]). fof(e2_87_3__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,0) = c1_87__prepower ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc1_rat_1,cc4_seqm_3,rc1_rat_1,rc2_rat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_seq_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_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_seq_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_87__prepower,dt_c4_87__prepower,dt_c5_87__prepower,de_c4_87__prepower,de_c5_87__prepower,fc2_membered,spc0_numerals,spc0_boole,e7_87__prepower]), [interesting(0.65),file(prepower,e2_87_3__prepower),[file(prepower,e2_87_3__prepower)]]). fof(e14_87__prepower,plain,( k2_seq_2(c5_87__prepower) = c1_87__prepower ), inference(iterative_eq,[status(thm),assumptions([dt_c1_87__prepower])],[e1_87_3__prepower,e2_87_3__prepower]), [interesting(0.8),file(prepower,e14_87__prepower),[file(prepower,e14_87__prepower)]]). fof(t33_seq_2,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( ( v4_seq_2(A) & v4_seq_2(B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) & k2_seq_2(A) = k2_seq_2(B) ) => v4_seq_2(C) ) ) ) ) ), file(seq_2,t33_seq_2), [interesting(0.9),axiom,file(seq_2,t33_seq_2)]). fof(e16_87__prepower,plain,( v4_seq_2(c3_87__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_ordinal2,fc4_seq_1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_rat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_zfmisc_1,dt_k4_seq_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_c2_87__prepower,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,rc2_rat_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_relset_1,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k10_seq_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_prepower,dt_k10_seq_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_seq_2,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_87__prepower,dt_c3_87__prepower,dt_c5_87__prepower,dt_c6_87__prepower,de_c3_87__prepower,de_c5_87__prepower,fc2_membered,e15_87__prepower,e11_87__prepower,e12_87__prepower,e13_87__prepower,e14_87__prepower,t33_seq_2]), [interesting(0.8),file(prepower,e16_87__prepower),[file(prepower,e16_87__prepower)]]). fof(t34_seq_2,theorem,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,k5_numbers,k1_numbers) & m2_relset_1(C,k5_numbers,k1_numbers) ) => ( ( v4_seq_2(A) & v4_seq_2(B) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,C,D)) & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) & k2_seq_2(A) = k2_seq_2(B) ) => k2_seq_2(C) = k2_seq_2(A) ) ) ) ) ), file(seq_2,t34_seq_2), [interesting(0.9),axiom,file(seq_2,t34_seq_2)]). fof(e17_87__prepower,plain,( k2_seq_2(c3_87__prepower) = c1_87__prepower ), inference(mizar_by,[status(thm),assumptions([dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_ordinal2,fc4_seq_1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_rat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,dt_k1_funct_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k2_zfmisc_1,dt_k4_seq_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_c2_87__prepower,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,rc2_rat_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_relset_1,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k10_seq_1,redefinition_k2_seq_1,redefinition_k2_seq_2,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_prepower,dt_k10_seq_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_seq_2,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_87__prepower,dt_c3_87__prepower,dt_c5_87__prepower,dt_c6_87__prepower,de_c3_87__prepower,de_c5_87__prepower,fc2_membered,e11_87__prepower,e12_87__prepower,e13_87__prepower,e14_87__prepower,e15_87__prepower,t34_seq_2]), [interesting(0.8),file(prepower,e17_87__prepower),[file(prepower,e17_87__prepower)]]). fof(dh_c8_87__prepower,definition, ( ( m2_subset_1(c8_87__prepower,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(c3_87__prepower,c8_87__prepower),c1_87__prepower) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(c3_87__prepower,A),c1_87__prepower) ) ), introduced(definition,[new_symbol(c8_87__prepower),file(prepower,c8_87__prepower)]), [interesting(0.8),axiom,file(prepower,c8_87__prepower)]). fof(dt_c8_87__prepower,assumption,( m2_subset_1(c8_87__prepower,k1_numbers,k5_numbers) ), introduced(assumption,[file(prepower,c8_87__prepower)]), [interesting(0.8),axiom,file(prepower,c8_87__prepower)]). fof(e18_87__prepower,plain,( r1_xreal_0(k10_prepower(c3_87__prepower,c8_87__prepower),k2_seq_1(k5_numbers,k1_numbers,c5_87__prepower,c8_87__prepower)) ), inference(mizar_by,[status(thm),assumptions([dt_c8_87__prepower,dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_c1_87__prepower,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_ordinal2,fc4_seq_1,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_rat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k4_seq_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c2_87__prepower,dt_c4_87__prepower,de_c4_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,rc2_rat_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k10_seq_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_prepower,dt_k10_seq_1,dt_k1_numbers,dt_k2_seq_1,dt_k5_numbers,dt_m2_subset_1,dt_c3_87__prepower,dt_c5_87__prepower,dt_c6_87__prepower,dt_c8_87__prepower,de_c3_87__prepower,de_c5_87__prepower,fc2_membered,e15_87__prepower]), [interesting(0.8),file(prepower,e18_87__prepower),[file(prepower,e18_87__prepower)]]). fof(e19_87__prepower,plain,( r1_xreal_0(k10_prepower(c3_87__prepower,c8_87__prepower),c1_87__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c8_87__prepower,dt_c1_87__prepower])],[cc4_seqm_3,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_seqm_3,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc4_ordinal2,fc6_membered,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_rat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t1_subset,t2_real,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_funcop_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c2_87__prepower,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc2_seqm_3,fc5_membered,rc2_rat_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k10_prepower,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k10_prepower,dt_k1_numbers,dt_k2_seq_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_87__prepower,dt_c3_87__prepower,dt_c4_87__prepower,dt_c5_87__prepower,dt_c8_87__prepower,de_c3_87__prepower,de_c4_87__prepower,de_c5_87__prepower,fc2_membered,e18_87__prepower,e7_87__prepower]), [interesting(0.8),file(prepower,e19_87__prepower),[file(prepower,e19_87__prepower)]]). fof(i6_87__prepower,theorem,( $true ), introduced(tautology,[file(prepower,i6_87__prepower)]), [interesting(0.8),trivial,file(prepower,i6_87__prepower)]). fof(i5_87__prepower,plain,( r1_xreal_0(k10_prepower(c3_87__prepower,c8_87__prepower),c1_87__prepower) ), inference(conclusion,[status(thm),assumptions([dt_c8_87__prepower,dt_c1_87__prepower])],[e19_87__prepower,i6_87__prepower]), [interesting(0.8),file(prepower,i5_87__prepower),[file(prepower,i5_87__prepower)]]). fof(i5_87_tmp__prepower,plain, ( m2_subset_1(c8_87__prepower,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(c3_87__prepower,c8_87__prepower),c1_87__prepower) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_87__prepower]),discharge_asm(discharge,[dt_c8_87__prepower])],[dt_c8_87__prepower,i5_87__prepower]), [interesting(0.8),i4_87__prepower]). fof(i4_87__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(c3_87__prepower,A),c1_87__prepower) ) ), inference(let,[status(thm),assumptions([dt_c1_87__prepower])],[i5_87_tmp__prepower,dh_c8_87__prepower]), [interesting(0.8),file(prepower,i4_87__prepower),[file(prepower,i4_87__prepower)]]). fof(i3_87__prepower,plain, ( k2_seq_2(c3_87__prepower) = c1_87__prepower & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(c3_87__prepower,A),c1_87__prepower) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_87__prepower])],[e17_87__prepower,i4_87__prepower]), [interesting(0.8),file(prepower,i3_87__prepower),[file(prepower,i3_87__prepower)]]). fof(i2_87__prepower,plain, ( v4_seq_2(c3_87__prepower) & k2_seq_2(c3_87__prepower) = c1_87__prepower & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(c3_87__prepower,A),c1_87__prepower) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_87__prepower])],[e16_87__prepower,i3_87__prepower]), [interesting(0.8),file(prepower,i2_87__prepower),[file(prepower,i2_87__prepower)]]). fof(i1_87__prepower,plain,( ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_prepower(A) & m2_relset_1(A,k5_numbers,k1_numbers) & v4_seq_2(A) & k2_seq_2(A) = c1_87__prepower & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(A,B),c1_87__prepower) ) ) ), inference(take,[status(thm),assumptions([dt_c1_87__prepower])],[cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_membered,cc2_rat_1,cc3_int_1,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,rc1_int_1,rc1_membered,rc1_nat_1,rc1_prepower,rc1_rat_1,rc1_seq_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,dt_k1_funct_1,dt_k1_seq_2,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_nat_1,cc1_rat_1,cc1_relset_1,cc1_seq_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc4_membered,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc2_rat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k10_prepower,redefinition_k2_seq_2,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_prepower,dt_k1_numbers,dt_k2_seq_2,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_87__prepower,dt_c3_87__prepower,fc2_membered,i2_87__prepower]), [interesting(0.8),file(prepower,i1_87__prepower),[file(prepower,i1_87__prepower)]]). fof(i1_87_tmp__prepower,plain, ( v1_xreal_0(c1_87__prepower) => ? [A] : ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & v1_prepower(A) & m2_relset_1(A,k5_numbers,k1_numbers) & v4_seq_2(A) & k2_seq_2(A) = c1_87__prepower & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(A,B),c1_87__prepower) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_87__prepower])],[dt_c1_87__prepower,i1_87__prepower]), [interesting(1),t79_prepower]). fof(t79_prepower,theorem,( ! [A] : ( v1_xreal_0(A) => ? [B] : ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & v1_prepower(B) & m2_relset_1(B,k5_numbers,k1_numbers) & v4_seq_2(B) & k2_seq_2(B) = A & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => r1_xreal_0(k10_prepower(B,C),A) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_87_tmp__prepower,dh_c1_87__prepower]), [interesting(1),file(prepower,t79_prepower),[file(prepower,t79_prepower)]]).