% Mizar ND problem: t4_prepower,prepower,113,57 fof(dh_c1_7__prepower,definition, ( ( v1_xreal_0(c1_7__prepower) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( A = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,A,0) = 1 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,B),c1_7__prepower) ) ) ) ) ) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,k5_numbers,k1_numbers) & m2_relset_1(D,k5_numbers,k1_numbers) ) => ( D = k2_prepower(C) <=> ( k2_seq_1(k5_numbers,k1_numbers,D,0) = 1 & ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,D,k1_nat_1(E,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,D,E),C) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_7__prepower),file(prepower,c1_7__prepower)]), [interesting(0.8),axiom,file(prepower,c1_7__prepower)]). fof(dh_c2_7__prepower,definition, ( ( ( v1_funct_1(c2_7__prepower) & v1_funct_2(c2_7__prepower,k5_numbers,k1_numbers) & m2_relset_1(c2_7__prepower,k5_numbers,k1_numbers) ) => ( c2_7__prepower = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ) ) ) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( B = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,B,0) = 1 & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,k1_nat_1(C,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,B,C),c1_7__prepower) ) ) ) ) ), introduced(definition,[new_symbol(c2_7__prepower),file(prepower,c2_7__prepower)]), [interesting(0.8),axiom,file(prepower,c2_7__prepower)]). fof(e1_7_1__prepower,assumption,( c2_7__prepower = k2_prepower(c1_7__prepower) ), introduced(assumption,[file(prepower,e1_7_1__prepower)]), [interesting(0.65),axiom,file(prepower,e1_7_1__prepower)]). 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(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(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_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(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc2_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_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), [interesting(0.9),axiom,file(int_1,cc4_int_1)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(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(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_int_1,theorem,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), [interesting(0.9),axiom,file(int_1,rc1_int_1)]). fof(rc1_membered,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ), file(membered,rc1_membered), [interesting(0.9),axiom,file(membered,rc1_membered)]). fof(rc1_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(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(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(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_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_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(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc1_newton,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v4_ordinal2(B) ) => ( v1_xcmplx_0(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) ) ) ), file(newton,fc1_newton), [interesting(0.9),axiom,file(newton,fc1_newton)]). 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(fc2_newton,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_newton(A,B)) & v1_xcmplx_0(k2_newton(A,B)) & v1_xreal_0(k2_newton(A,B)) & ~ v3_xreal_0(k2_newton(A,B)) & v1_int_1(k2_newton(A,B)) ) ) ), file(newton,fc2_newton), [interesting(0.9),axiom,file(newton,fc2_newton)]). fof(fc5_membered,theorem, ( v1_membered(k5_ordinal2) & v2_membered(k5_ordinal2) & v3_membered(k5_ordinal2) & v4_membered(k5_ordinal2) & v5_membered(k5_ordinal2) ), file(membered,fc5_membered), [interesting(0.9),axiom,file(membered,fc5_membered)]). fof(rc1_nat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(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(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k2_seq_1,definition,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(redefinition_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_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k2_newton,axiom,( $true ), file(newton,k2_newton), [interesting(0.9),axiom,file(newton,k2_newton)]). fof(dt_k2_prepower,axiom,( ! [A] : ( v1_xreal_0(A) => ( v1_funct_1(k2_prepower(A)) & v1_funct_2(k2_prepower(A),k5_numbers,k1_numbers) & m2_relset_1(k2_prepower(A),k5_numbers,k1_numbers) ) ) ), file(prepower,k2_prepower), [interesting(0.9),axiom,file(prepower,k2_prepower)]). 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_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_7__prepower,assumption,( v1_xreal_0(c1_7__prepower) ), introduced(assumption,[file(prepower,c1_7__prepower)]), [interesting(0.8),axiom,file(prepower,c1_7__prepower)]). fof(dt_c2_7__prepower,assumption, ( v1_funct_1(c2_7__prepower) & v1_funct_2(c2_7__prepower,k5_numbers,k1_numbers) & m2_relset_1(c2_7__prepower,k5_numbers,k1_numbers) ), introduced(assumption,[file(prepower,c2_7__prepower)]), [interesting(0.8),axiom,file(prepower,c2_7__prepower)]). 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(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(d1_prepower,definition,( ! [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) ) => ( B = k2_prepower(A) <=> ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_newton(A,C) ) ) ) ) ), file(prepower,d1_prepower), [interesting(0.9),axiom,file(prepower,d1_prepower)]). fof(e1_7_1_1__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = k2_newton(c1_7__prepower,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[cc1_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_seq_1,fc6_membered,rc1_int_1,rc1_membered,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_newton,fc1_ordinal2,fc2_newton,fc5_membered,rc1_nat_1,rc1_xreal_0,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_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_newton,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c2_7__prepower,cc2_xreal_0,fc2_membered,spc0_numerals,spc0_boole,e1_7_1__prepower,d1_prepower]), [interesting(0.5),file(prepower,e1_7_1_1__prepower),[file(prepower,e1_7_1_1__prepower)]]). 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(t9_newton,theorem,( ! [A] : ( v1_xreal_0(A) => k2_newton(A,0) = 1 ) ), file(newton,t9_newton), [interesting(0.9),axiom,file(newton,t9_newton)]). fof(e2_7_1_1__prepower,plain,( k2_newton(c1_7__prepower,0) = 1 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower])],[reflexivity_r1_tarski,cc1_rat_1,rc1_rat_1,rc2_rat_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,fc6_membered,rc1_int_1,rc1_membered,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_subset,t3_subset,t4_subset,t5_subset,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_newton,fc2_membered,fc2_newton,rc1_nat_1,rc1_xreal_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,dt_k2_newton,dt_c1_7__prepower,cc2_xreal_0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t9_newton]), [interesting(0.5),file(prepower,e2_7_1_1__prepower),[file(prepower,e2_7_1_1__prepower)]]). fof(e2_7_1__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 ), inference(iterative_eq,[status(thm),assumptions([dt_c2_7__prepower,e1_7_1__prepower,dt_c1_7__prepower])],[e1_7_1_1__prepower,e2_7_1_1__prepower]), [interesting(0.65),file(prepower,e2_7_1__prepower),[file(prepower,e2_7_1__prepower)]]). fof(dh_c1_7_1__prepower,definition, ( ( m2_subset_1(c1_7_1__prepower,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_1__prepower,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_1__prepower),c1_7__prepower) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ), introduced(definition,[new_symbol(c1_7_1__prepower),file(prepower,c1_7_1__prepower)]), [interesting(0.65),axiom,file(prepower,c1_7_1__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(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(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(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(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(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc1_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), [interesting(0.9),axiom,file(int_1,fc1_int_1)]). fof(fc6_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), [interesting(0.9),axiom,file(int_1,fc6_int_1)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(commutativity_k2_xcmplx_0,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,B) = k2_xcmplx_0(B,A) ) ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(fc1_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), [interesting(0.9),axiom,file(nat_1,fc1_nat_1)]). fof(fc3_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), [interesting(0.9),axiom,file(nat_1,fc3_nat_1)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(fc4_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), [interesting(0.9),axiom,file(nat_1,fc4_nat_1)]). fof(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(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(commutativity_k1_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(redefinition_k1_nat_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_c1_7_1__prepower,assumption,( m2_subset_1(c1_7_1__prepower,k1_numbers,k5_numbers) ), introduced(assumption,[file(prepower,c1_7_1__prepower)]), [interesting(0.65),axiom,file(prepower,c1_7_1__prepower)]). fof(e1_7_1_2__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_1__prepower,1)) = k2_newton(c1_7__prepower,k1_nat_1(c1_7_1__prepower,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_1__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[cc1_rat_1,fc10_rat_1,fc13_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_seq_1,fc6_int_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,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,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_nat_1,fc1_newton,fc1_ordinal2,fc2_newton,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_membered,fc8_xreal_0,rc1_nat_1,rc1_xreal_0,spc6_arithm,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_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_newton,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c1_7_1__prepower,dt_c2_7__prepower,cc2_xreal_0,fc2_membered,spc1_numerals,spc1_boole,e1_7_1__prepower,d1_prepower]), [interesting(0.5),file(prepower,e1_7_1_2__prepower),[file(prepower,e1_7_1_2__prepower)]]). 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(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(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(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(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(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(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(fc24_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc24_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc24_xreal_0)]). fof(fc2_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k3_xcmplx_0(A,B)) & v1_xreal_0(k3_xcmplx_0(A,B)) & v1_int_1(k3_xcmplx_0(A,B)) ) ) ), file(int_1,fc2_int_1), [interesting(0.9),axiom,file(int_1,fc2_int_1)]). fof(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(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(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(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(dt_k3_xcmplx_0,axiom,( $true ), file(xcmplx_0,k3_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k3_xcmplx_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(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(t11_newton,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_xreal_0(B) => k2_newton(B,k2_xcmplx_0(A,1)) = k3_xcmplx_0(k2_newton(B,A),B) ) ) ), file(newton,t11_newton), [interesting(0.9),axiom,file(newton,t11_newton)]). 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(e2_7_1_2__prepower,plain,( k2_newton(c1_7__prepower,k1_nat_1(c1_7_1__prepower,1)) = k3_xcmplx_0(k2_newton(c1_7__prepower,c1_7_1__prepower),c1_7__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_1__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,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc10_rat_1,fc12_rat_1,fc13_rat_1,fc15_rat_1,fc1_ordinal2,fc1_rat_1,fc2_rat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc9_rat_1,rc1_membered,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,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,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,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc2_int_1,fc2_membered,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc5_arithm,spc6_arithm,spc7_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_newton,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_c1_7__prepower,dt_c1_7_1__prepower,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,fc1_newton,fc2_nat_1,fc2_newton,fc3_xreal_0,fc4_xreal_0,spc1_numerals,spc1_boole,t11_newton,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(prepower,e2_7_1_2__prepower),[file(prepower,e2_7_1_2__prepower)]]). fof(e3_7_1_2__prepower,plain,( k3_xcmplx_0(k2_newton(c1_7__prepower,c1_7_1__prepower),c1_7__prepower) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_1__prepower),c1_7__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_1__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[cc1_rat_1,fc12_rat_1,fc15_rat_1,fc1_rat_1,fc6_rat_1,fc9_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_newton,fc1_ordinal2,fc23_xreal_0,fc2_nat_1,fc2_newton,fc5_membered,rc1_nat_1,rc1_xreal_0,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xcmplx_0,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_newton,dt_k2_prepower,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c1_7_1__prepower,dt_c2_7__prepower,cc2_xreal_0,fc2_membered,fc4_xreal_0,spc1_numerals,spc1_boole,e1_7_1__prepower,d1_prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(prepower,e3_7_1_2__prepower),[file(prepower,e3_7_1_2__prepower)]]). fof(e3_7_1__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_1__prepower,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_1__prepower),c1_7__prepower) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_1__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[e1_7_1_2__prepower,e2_7_1_2__prepower,e3_7_1_2__prepower]), [interesting(0.65),file(prepower,e3_7_1__prepower),[file(prepower,e3_7_1__prepower)]]). fof(i4_7_1__prepower,theorem,( $true ), introduced(tautology,[file(prepower,i4_7_1__prepower)]), [interesting(0.65),trivial,file(prepower,i4_7_1__prepower)]). fof(i3_7_1__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_1__prepower,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_1__prepower),c1_7__prepower) ), inference(conclusion,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_1__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[e3_7_1__prepower,i4_7_1__prepower]), [interesting(0.65),file(prepower,i3_7_1__prepower),[file(prepower,i3_7_1__prepower)]]). fof(i3_7_1_tmp__prepower,plain, ( m2_subset_1(c1_7_1__prepower,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_1__prepower,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_1__prepower),c1_7__prepower) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__prepower,dt_c2_7__prepower,e1_7_1__prepower]),discharge_asm(discharge,[dt_c1_7_1__prepower])],[dt_c1_7_1__prepower,i3_7_1__prepower]), [interesting(0.65),i2_7_1__prepower]). fof(i2_7_1__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ), inference(let,[status(thm),assumptions([dt_c1_7__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[i3_7_1_tmp__prepower,dh_c1_7_1__prepower]), [interesting(0.65),file(prepower,i2_7_1__prepower),[file(prepower,i2_7_1__prepower)]]). fof(i1_7_1__prepower,plain, ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_7__prepower,dt_c2_7__prepower,e1_7_1__prepower])],[e2_7_1__prepower,i2_7_1__prepower]), [interesting(0.65),file(prepower,i1_7_1__prepower),[file(prepower,i1_7_1__prepower)]]). fof(e1_7__prepower,plain, ( c2_7__prepower = k2_prepower(c1_7__prepower) => ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__prepower,dt_c2_7__prepower]),discharge_asm(discharge,[e1_7_1__prepower])],[e1_7_1__prepower,i1_7_1__prepower]), [interesting(0.8),file(prepower,e1_7__prepower),[file(prepower,e1_7__prepower)]]). fof(e2_7__prepower,assumption,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 ), introduced(assumption,[file(prepower,e2_7__prepower)]), [interesting(0.8),axiom,file(prepower,e2_7__prepower)]). fof(e3_7__prepower,assumption,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ), introduced(assumption,[file(prepower,e3_7__prepower)]), [interesting(0.8),axiom,file(prepower,e3_7__prepower)]). fof(s1_nat_1__e7_7__prepower,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_funct_1(B) & v1_funct_2(B,k5_numbers,k1_numbers) & m2_relset_1(B,k5_numbers,k1_numbers) ) => ( ( k2_seq_1(k5_numbers,k1_numbers,B,0) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(A),0) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(A),C) => k2_seq_1(k5_numbers,k1_numbers,B,k1_nat_1(C,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(A),k1_nat_1(C,1)) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,C) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(A),C) ) ) ) ), file(prepower,s1_nat_1__e7_7__prepower), [interesting(0.9),axiom,file(prepower,s1_nat_1__e7_7__prepower)]). fof(e1_7_2__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = k2_newton(c1_7__prepower,0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c2_7__prepower,e2_7__prepower])],[cc1_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_seq_1,fc6_membered,rc1_int_1,rc1_membered,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,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_zfmisc_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_seq_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_newton,fc1_ordinal2,fc2_newton,fc5_membered,rc1_nat_1,rc1_xreal_0,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,redefinition_k2_seq_1,redefinition_k5_numbers,dt_k1_numbers,dt_k2_newton,dt_k2_seq_1,dt_k5_numbers,dt_c1_7__prepower,dt_c2_7__prepower,cc2_xreal_0,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e2_7__prepower,t9_newton]), [interesting(0.65),file(prepower,e1_7_2__prepower),[file(prepower,e1_7_2__prepower)]]). fof(e2_7_2__prepower,plain,( k2_newton(c1_7__prepower,0) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower])],[cc1_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_seq_1,fc6_membered,rc1_int_1,rc1_membered,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_newton,fc1_ordinal2,fc2_newton,fc5_membered,rc1_nat_1,rc1_xreal_0,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_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_newton,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,cc2_xreal_0,fc2_membered,spc0_numerals,spc0_boole,d1_prepower]), [interesting(0.65),file(prepower,e2_7_2__prepower),[file(prepower,e2_7_2__prepower)]]). fof(e4_7__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),0) ), inference(iterative_eq,[status(thm),assumptions([dt_c2_7__prepower,e2_7__prepower,dt_c1_7__prepower])],[e1_7_2__prepower,e2_7_2__prepower]), [interesting(0.8),file(prepower,e4_7__prepower),[file(prepower,e4_7__prepower)]]). fof(e5_7__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),0) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7__prepower,e2_7__prepower,dt_c1_7__prepower])],[cc1_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,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_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_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_zfmisc_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_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,fc5_membered,t1_numerals,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,redefinition_k2_seq_1,redefinition_k5_numbers,dt_k1_numbers,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_c1_7__prepower,dt_c2_7__prepower,fc2_membered,spc0_numerals,spc0_boole,e4_7__prepower]), [interesting(0.8),file(prepower,e5_7__prepower),[file(prepower,e5_7__prepower)]]). fof(dh_c1_7_3__prepower,definition, ( ( m2_subset_1(c1_7_3__prepower,k1_numbers,k5_numbers) => ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_3__prepower) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),c1_7_3__prepower) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(c1_7_3__prepower,1)) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),A) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(A,1)) ) ) ), introduced(definition,[new_symbol(c1_7_3__prepower),file(prepower,c1_7_3__prepower)]), [interesting(0.65),axiom,file(prepower,c1_7_3__prepower)]). fof(e1_7_3__prepower,assumption,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_3__prepower) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),c1_7_3__prepower) ), introduced(assumption,[file(prepower,e1_7_3__prepower)]), [interesting(0.65),axiom,file(prepower,e1_7_3__prepower)]). fof(dt_c1_7_3__prepower,assumption,( m2_subset_1(c1_7_3__prepower,k1_numbers,k5_numbers) ), introduced(assumption,[file(prepower,c1_7_3__prepower)]), [interesting(0.65),axiom,file(prepower,c1_7_3__prepower)]). fof(e1_7_3_1__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_3__prepower),c1_7__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_3__prepower,dt_c2_7__prepower,e3_7__prepower])],[cc1_rat_1,fc10_rat_1,fc12_rat_1,fc13_rat_1,fc15_rat_1,fc1_rat_1,fc2_rat_1,fc4_rat_1,fc6_rat_1,fc7_rat_1,fc9_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,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc2_nat_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc6_membered,fc7_int_1,fc7_xreal_0,fc9_xreal_0,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,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_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc23_xreal_0,fc3_xreal_0,fc4_xreal_0,fc5_membered,fc8_xreal_0,rc1_xreal_0,spc5_arithm,spc6_arithm,spc7_arithm,t2_subset,t3_arithm,t3_subset,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_nat_1,dt_k1_numbers,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__prepower,dt_c1_7_3__prepower,dt_c2_7__prepower,fc2_membered,spc1_numerals,spc1_boole,e3_7__prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(prepower,e1_7_3_1__prepower),[file(prepower,e1_7_3_1__prepower)]]). fof(e2_7_3_1__prepower,plain,( k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_3__prepower),c1_7__prepower) = k3_xcmplx_0(k2_newton(c1_7__prepower,c1_7_3__prepower),c1_7__prepower) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_3__prepower,dt_c2_7__prepower,e1_7_3__prepower])],[cc1_rat_1,fc12_rat_1,fc15_rat_1,fc1_rat_1,fc6_rat_1,fc9_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_seq_1,fc21_xreal_0,fc22_xreal_0,fc24_xreal_0,fc2_int_1,fc6_membered,fc7_int_1,rc1_int_1,rc1_membered,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_newton,fc1_ordinal2,fc23_xreal_0,fc2_nat_1,fc2_newton,fc5_membered,rc1_nat_1,rc1_xreal_0,spc7_arithm,t2_subset,t3_arithm,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_xcmplx_0,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_newton,dt_k2_prepower,dt_k2_seq_1,dt_k3_xcmplx_0,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c1_7_3__prepower,dt_c2_7__prepower,cc2_xreal_0,fc2_membered,fc4_xreal_0,spc1_numerals,spc1_boole,e1_7_3__prepower,d1_prepower,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(prepower,e2_7_3_1__prepower),[file(prepower,e2_7_3_1__prepower)]]). fof(e3_7_3_1__prepower,plain,( k3_xcmplx_0(k2_newton(c1_7__prepower,c1_7_3__prepower),c1_7__prepower) = k2_newton(c1_7__prepower,k1_nat_1(c1_7_3__prepower,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_3__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,cc20_membered,cc2_membered,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc10_rat_1,fc12_rat_1,fc13_rat_1,fc15_rat_1,fc1_ordinal2,fc1_rat_1,fc2_rat_1,fc4_rat_1,fc5_membered,fc6_membered,fc6_rat_1,fc7_rat_1,fc9_rat_1,rc1_membered,rc1_rat_1,rc2_nat_1,rc2_rat_1,rc3_nat_1,t1_subset,t3_subset,t4_subset,t5_subset,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,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,fc21_xreal_0,fc22_xreal_0,fc23_xreal_0,fc24_xreal_0,fc2_int_1,fc2_membered,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc5_arithm,spc6_arithm,spc7_arithm,t2_subset,t3_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,commutativity_k3_xcmplx_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_newton,dt_k2_xcmplx_0,dt_k3_xcmplx_0,dt_c1_7__prepower,dt_c1_7_3__prepower,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,fc1_newton,fc2_nat_1,fc2_newton,fc3_xreal_0,fc4_xreal_0,spc1_numerals,spc1_boole,t11_newton,rqRealMult__k3_xcmplx_0__r1_r1_r1]), [interesting(0.5),file(prepower,e3_7_3_1__prepower),[file(prepower,e3_7_3_1__prepower)]]). fof(e4_7_3_1__prepower,plain,( k2_newton(c1_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(c1_7_3__prepower,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__prepower,dt_c1_7_3__prepower])],[cc1_rat_1,fc10_rat_1,fc13_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,cc20_membered,cc2_membered,cc2_rat_1,cc3_membered,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_seq_1,fc6_int_1,fc6_membered,fc7_xreal_0,fc9_xreal_0,rc1_int_1,rc1_membered,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,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,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,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_nat_1,fc1_newton,fc1_ordinal2,fc2_newton,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_membered,fc8_xreal_0,rc1_nat_1,rc1_xreal_0,spc6_arithm,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_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_newton,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c1_7_3__prepower,cc2_xreal_0,fc2_membered,spc1_numerals,spc1_boole,d1_prepower]), [interesting(0.5),file(prepower,e4_7_3_1__prepower),[file(prepower,e4_7_3_1__prepower)]]). fof(e2_7_3__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(c1_7_3__prepower,1)) ), inference(iterative_eq,[status(thm),assumptions([e3_7__prepower,dt_c2_7__prepower,e1_7_3__prepower,dt_c1_7__prepower,dt_c1_7_3__prepower])],[e1_7_3_1__prepower,e2_7_3_1__prepower,e3_7_3_1__prepower,e4_7_3_1__prepower]), [interesting(0.65),file(prepower,e2_7_3__prepower),[file(prepower,e2_7_3__prepower)]]). fof(i3_7_3__prepower,theorem,( $true ), introduced(tautology,[file(prepower,i3_7_3__prepower)]), [interesting(0.65),trivial,file(prepower,i3_7_3__prepower)]). fof(i2_7_3__prepower,plain,( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(c1_7_3__prepower,1)) ), inference(conclusion,[status(thm),assumptions([e3_7__prepower,dt_c2_7__prepower,e1_7_3__prepower,dt_c1_7__prepower,dt_c1_7_3__prepower])],[e2_7_3__prepower,i3_7_3__prepower]), [interesting(0.65),file(prepower,i2_7_3__prepower),[file(prepower,i2_7_3__prepower)]]). fof(i1_7_3__prepower,plain, ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_3__prepower) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),c1_7_3__prepower) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(c1_7_3__prepower,1)) ), inference(discharge_asm,[status(thm),assumptions([e3_7__prepower,dt_c2_7__prepower,dt_c1_7__prepower,dt_c1_7_3__prepower]),discharge_asm(discharge,[e1_7_3__prepower])],[e1_7_3__prepower,i2_7_3__prepower]), [interesting(0.65),file(prepower,i1_7_3__prepower),[file(prepower,i1_7_3__prepower)]]). fof(i1_7_3_tmp__prepower,plain, ( m2_subset_1(c1_7_3__prepower,k1_numbers,k5_numbers) => ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,c1_7_3__prepower) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),c1_7_3__prepower) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(c1_7_3__prepower,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(c1_7_3__prepower,1)) ) ), inference(discharge_asm,[status(thm),assumptions([e3_7__prepower,dt_c2_7__prepower,dt_c1_7__prepower]),discharge_asm(discharge,[dt_c1_7_3__prepower])],[dt_c1_7_3__prepower,i1_7_3__prepower]), [interesting(0.8),e6_7__prepower]). fof(e6_7__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),A) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),k1_nat_1(A,1)) ) ) ), inference(let,[status(thm),assumptions([e3_7__prepower,dt_c2_7__prepower,dt_c1_7__prepower])],[i1_7_3_tmp__prepower,dh_c1_7_3__prepower]), [interesting(0.8),file(prepower,e6_7__prepower),[file(prepower,e6_7__prepower)]]). fof(e7_7__prepower,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A) = k2_seq_1(k5_numbers,k1_numbers,k2_prepower(c1_7__prepower),A) ) ), inference(mizar_from,[status(thm),assumptions([e2_7__prepower,e3_7__prepower,dt_c2_7__prepower,dt_c1_7__prepower])],[cc1_rat_1,fc10_rat_1,fc13_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,cc3_int_1,cc3_membered,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_seq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,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,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,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc3_xreal_0,fc5_membered,fc8_xreal_0,rc1_xreal_0,commutativity_k1_nat_1,redefinition_k1_nat_1,redefinition_k2_seq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c2_7__prepower,cc2_xreal_0,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,s1_nat_1__e7_7__prepower,e5_7__prepower,e6_7__prepower]), [interesting(0.8),file(prepower,e7_7__prepower),[file(prepower,e7_7__prepower)]]). fof(t113_funct_2,theorem,( ! [A,B,C] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m2_relset_1(C,A,B) ) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( ! [E] : ( m1_subset_1(E,A) => k1_funct_1(C,E) = k1_funct_1(D,E) ) => C = D ) ) ) ), file(funct_2,t113_funct_2), [interesting(0.9),axiom,file(funct_2,t113_funct_2)]). fof(e8_7__prepower,plain,( c2_7__prepower = k2_prepower(c1_7__prepower) ), inference(mizar_by,[status(thm),assumptions([e2_7__prepower,e3_7__prepower,dt_c2_7__prepower,dt_c1_7__prepower])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_rat_1,cc20_membered,cc2_membered,cc3_membered,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,fc1_seq_1,fc6_membered,rc1_membered,rc1_rat_1,rc1_seq_1,rc2_rat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_relset_1,cc1_seq_1,cc1_xreal_0,cc2_rat_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc1_ordinal2,fc5_membered,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,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_funct_1,dt_k1_numbers,dt_k2_prepower,dt_k2_seq_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_7__prepower,dt_c2_7__prepower,cc1_nat_1,cc2_int_1,cc2_nat_1,fc2_membered,e7_7__prepower,t113_funct_2]), [interesting(0.8),file(prepower,e8_7__prepower),[file(prepower,e8_7__prepower)]]). fof(i5_7__prepower,theorem,( $true ), introduced(tautology,[file(prepower,i5_7__prepower)]), [interesting(0.8),trivial,file(prepower,i5_7__prepower)]). fof(i4_7__prepower,plain,( c2_7__prepower = k2_prepower(c1_7__prepower) ), inference(conclusion,[status(thm),assumptions([e2_7__prepower,e3_7__prepower,dt_c2_7__prepower,dt_c1_7__prepower])],[e8_7__prepower,i5_7__prepower]), [interesting(0.8),file(prepower,i4_7__prepower),[file(prepower,i4_7__prepower)]]). fof(i3_7__prepower,plain, ( ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ) => c2_7__prepower = k2_prepower(c1_7__prepower) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_7__prepower,dt_c1_7__prepower]),discharge_asm(discharge,[e2_7__prepower,e3_7__prepower])],[e2_7__prepower,e3_7__prepower,i4_7__prepower]), [interesting(0.8),file(prepower,i3_7__prepower),[file(prepower,i3_7__prepower)]]). fof(i2_7__prepower,plain, ( c2_7__prepower = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c2_7__prepower,dt_c1_7__prepower])],[e1_7__prepower,i3_7__prepower]), [interesting(0.8),file(prepower,i2_7__prepower),[file(prepower,i2_7__prepower)]]). fof(i2_7_tmp__prepower,plain, ( ( v1_funct_1(c2_7__prepower) & v1_funct_2(c2_7__prepower,k5_numbers,k1_numbers) & m2_relset_1(c2_7__prepower,k5_numbers,k1_numbers) ) => ( c2_7__prepower = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,0) = 1 & ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,k1_nat_1(A,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,c2_7__prepower,A),c1_7__prepower) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__prepower]),discharge_asm(discharge,[dt_c2_7__prepower])],[dt_c2_7__prepower,i2_7__prepower]), [interesting(0.8),i1_7__prepower]). fof(i1_7__prepower,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( A = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,A,0) = 1 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,B),c1_7__prepower) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_7__prepower])],[i2_7_tmp__prepower,dh_c2_7__prepower]), [interesting(0.8),file(prepower,i1_7__prepower),[file(prepower,i1_7__prepower)]]). fof(i1_7_tmp__prepower,plain, ( v1_xreal_0(c1_7__prepower) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,k5_numbers,k1_numbers) & m2_relset_1(A,k5_numbers,k1_numbers) ) => ( A = k2_prepower(c1_7__prepower) <=> ( k2_seq_1(k5_numbers,k1_numbers,A,0) = 1 & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(B,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,B),c1_7__prepower) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_7__prepower])],[dt_c1_7__prepower,i1_7__prepower]), [interesting(1),t4_prepower]). fof(t4_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) ) => ( B = k2_prepower(A) <=> ( k2_seq_1(k5_numbers,k1_numbers,B,0) = 1 & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => k2_seq_1(k5_numbers,k1_numbers,B,k1_nat_1(C,1)) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,B,C),A) ) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_7_tmp__prepower,dh_c1_7__prepower]), [interesting(1),file(prepower,t4_prepower),[file(prepower,t4_prepower)]]).