% Mizar ND problem: t5_calcul_2,calcul_2,108,32 fof(dh_c1_7__calcul_2,definition, ( ( v4_ordinal2(c1_7__calcul_2) => ! [A] : ( v4_ordinal2(A) => k2_xboole_0(k2_calcul_2(c1_7__calcul_2,A),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,A),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(A,1)) ) ) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => k2_xboole_0(k2_calcul_2(B,C),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(B,C),1))) = k2_calcul_2(B,k2_xcmplx_0(C,1)) ) ) ), introduced(definition,[new_symbol(c1_7__calcul_2),file(calcul_2,c1_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c1_7__calcul_2)]). fof(dh_c2_7__calcul_2,definition, ( ( v4_ordinal2(c2_7__calcul_2) => k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1)) ) => ! [A] : ( v4_ordinal2(A) => k2_xboole_0(k2_calcul_2(c1_7__calcul_2,A),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,A),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(A,1)) ) ), introduced(definition,[new_symbol(c2_7__calcul_2),file(calcul_2,c2_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c2_7__calcul_2)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_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(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(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(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc11_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc11_xreal_0)]). fof(fc12_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & ~ v2_xreal_0(k2_xcmplx_0(B,A)) & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc12_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc12_xreal_0)]). fof(fc1_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), [interesting(0.9),axiom,file(int_1,fc1_int_1)]). fof(fc1_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(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_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(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_nat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_int_1,theorem,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), [interesting(0.9),axiom,file(int_1,rc2_int_1)]). fof(rc2_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_1)]). fof(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(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k1_calcul_2,axiom,( $true ), file(calcul_2,k1_calcul_2), [interesting(0.9),axiom,file(calcul_2,k1_calcul_2)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k5_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_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(cc1_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(cc1_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), [interesting(0.9),axiom,file(nat_1,cc1_nat_1)]). fof(cc1_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc2_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(fc1_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), [interesting(0.9),axiom,file(nat_1,fc1_nat_1)]). fof(fc3_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), [interesting(0.9),axiom,file(nat_1,fc3_nat_1)]). fof(fc4_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), [interesting(0.9),axiom,file(nat_1,fc4_nat_1)]). fof(commutativity_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]). fof(idempotence_k2_xboole_0,theorem,( ! [A,B] : k2_xboole_0(A,A) = A ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_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(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(redefinition_k2_calcul_2,definition,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => k2_calcul_2(A,B) = k1_calcul_2(A,B) ) ), file(calcul_2,k2_calcul_2), [interesting(0.9),axiom,file(calcul_2,k2_calcul_2)]). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). fof(dt_k2_calcul_2,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m1_subset_1(k2_calcul_2(A,B),k1_zfmisc_1(k5_numbers)) ) ), file(calcul_2,k2_calcul_2), [interesting(0.9),axiom,file(calcul_2,k2_calcul_2)]). fof(dt_k2_xboole_0,axiom,( $true ), file(xboole_0,k2_xboole_0), [interesting(0.9),axiom,file(xboole_0,k2_xboole_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(dt_c1_7__calcul_2,assumption,( v4_ordinal2(c1_7__calcul_2) ), introduced(assumption,[file(calcul_2,c1_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c1_7__calcul_2)]). fof(dt_c2_7__calcul_2,assumption,( v4_ordinal2(c2_7__calcul_2) ), introduced(assumption,[file(calcul_2,c2_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c2_7__calcul_2)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]). 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_c1_7_1__calcul_2,assumption,( $true ), introduced(assumption,[file(calcul_2,c1_7_1__calcul_2)]), [interesting(0.65),axiom,file(calcul_2,c1_7_1__calcul_2)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(dh_c1_7_1__calcul_2,definition, ( ~ ( r2_hidden(c1_7_1__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) & ~ r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ) => ! [A] : ~ ( r2_hidden(A,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) & ~ r2_hidden(A,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ) ), introduced(definition,[new_symbol(c1_7_1__calcul_2),file(calcul_2,c1_7_1__calcul_2)]), [interesting(0.65),axiom,file(calcul_2,c1_7_1__calcul_2)]). fof(e3_7_1__calcul_2,assumption,( r2_hidden(c1_7_1__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ), introduced(assumption,[file(calcul_2,e3_7_1__calcul_2)]), [interesting(0.65),axiom,file(calcul_2,e3_7_1__calcul_2)]). 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(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(cc2_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc2_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(rc4_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(rc8_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). 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__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_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_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(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_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(fc2_finseq_1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_relat_1(k1_xboole_0) & v3_relat_1(k1_xboole_0) & v1_funct_1(k1_xboole_0) & v2_funct_1(k1_xboole_0) & v1_finset_1(k1_xboole_0) & v1_finseq_1(k1_xboole_0) ), file(finseq_1,fc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc2_finseq_1)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_2_0_calcul_2,definition,( ! [A,B,C] : ( ( v4_ordinal2(B) & v4_ordinal2(C) ) => ( r2_hidden(A,a_2_0_calcul_2(B,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = D & r1_xreal_0(k2_xcmplx_0(1,B),D) & r1_xreal_0(D,k2_xcmplx_0(C,B)) ) ) ) ), file(calcul_2,a_2_0_calcul_2), [interesting(0.9),axiom,file(calcul_2,a_2_0_calcul_2)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(existence_m2_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(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_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(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(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(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(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(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(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_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(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(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(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(t1_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(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(d1_calcul_2,definition,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => k1_calcul_2(A,B) = a_2_0_calcul_2(A,B) ) ) ), file(calcul_2,d1_calcul_2), [interesting(0.9),axiom,file(calcul_2,d1_calcul_2)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(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(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_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__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(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(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(t1_boole,theorem,( ! [A] : k2_xboole_0(A,k1_xboole_0) = A ), file(boole,t1_boole), [interesting(0.9),axiom,file(boole,t1_boole)]). fof(d2_xboole_0,definition,( ! [A,B,C] : ( C = k2_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) | r2_hidden(D,B) ) ) ) ), file(xboole_0,d2_xboole_0), [interesting(0.9),axiom,file(xboole_0,d2_xboole_0)]). fof(e4_7_1__calcul_2,plain, ( r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2)) | r2_hidden(c1_7_1__calcul_2,k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c1_7_1__calcul_2,dt_c2_7__calcul_2,e3_7_1__calcul_2])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,cc2_xreal_0,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_ordinal2,fc2_finseq_1,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_boole,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,spc6_arithm,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_calcul_2,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c1_7_1__calcul_2,dt_c2_7__calcul_2,t1_subset,t7_boole,spc1_numerals,spc1_boole,e3_7_1__calcul_2,d2_xboole_0]), [interesting(0.65),file(calcul_2,e4_7_1__calcul_2),[file(calcul_2,e4_7_1__calcul_2)]]). fof(t9_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), [interesting(0.9),axiom,file(xreal_1,t9_xreal_1)]). fof(e1_7_1__calcul_2,plain,( r1_xreal_0(k2_xcmplx_0(c2_7__calcul_2,0),k2_xcmplx_0(c2_7__calcul_2,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7__calcul_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,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_ordinal2,fc2_finseq_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_int_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,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,cc1_funct_1,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_nat_1,fc3_nat_1,fc4_nat_1,fc8_xreal_0,rc1_nat_1,rc1_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k2_xcmplx_0,dt_c2_7__calcul_2,cc2_xreal_0,fc3_xreal_0,rqLessOrEqual__r1_xreal_0__r1_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t9_xreal_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(calcul_2,e1_7_1__calcul_2),[file(calcul_2,e1_7_1__calcul_2)]]). fof(t4_calcul_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( r1_xreal_0(A,B) <=> r1_tarski(k2_calcul_2(C,A),k2_calcul_2(C,B)) ) ) ) ) ), file(calcul_2,t4_calcul_2), [interesting(0.9),axiom,file(calcul_2,t4_calcul_2)]). fof(e2_7_1__calcul_2,plain,( r1_tarski(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k5_ordinal2,fc1_ordinal2,fc2_finseq_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,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,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_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,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,d1_calcul_2,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_calcul_2,dt_k2_calcul_2,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r1_r0,t3_subset,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_7_1__calcul_2,t4_calcul_2,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(calcul_2,e2_7_1__calcul_2),[file(calcul_2,e2_7_1__calcul_2)]]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.9),axiom,file(tarski,d1_tarski)]). fof(e5_7_1__calcul_2,plain, ( r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) | c1_7_1__calcul_2 = k2_xcmplx_0(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_1__calcul_2,e3_7_1__calcul_2,dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,dt_k1_xboole_0,dt_k5_ordinal2,cc2_xreal_0,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_ordinal2,fc2_finseq_1,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,spc6_arithm,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_calcul_2,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c1_7_1__calcul_2,dt_c2_7__calcul_2,t1_subset,t3_subset,t7_boole,spc1_numerals,spc1_boole,e4_7_1__calcul_2,e2_7_1__calcul_2,d1_tarski]), [interesting(0.65),file(calcul_2,e5_7_1__calcul_2),[file(calcul_2,e5_7_1__calcul_2)]]). fof(t3_calcul_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( A = 0 | r2_hidden(k2_xcmplx_0(A,B),k2_calcul_2(B,A)) ) ) ) ), file(calcul_2,t3_calcul_2), [interesting(0.9),axiom,file(calcul_2,t3_calcul_2)]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(rqRealDiff__k6_xcmplx_0__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__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(e6_7_1__calcul_2,plain,( r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_1__calcul_2,e3_7_1__calcul_2,dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,fc1_ordinal2,fc2_finseq_1,rc2_nat_1,rc3_nat_1,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,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,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_xreal_0,fc20_xreal_0,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_int_1,fc5_xreal_0,fc6_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t4_subset,t5_subset,t6_boole,t8_boole,d1_calcul_2,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k2_calcul_2,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__calcul_2,dt_c1_7_1__calcul_2,dt_c2_7__calcul_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_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_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,t1_subset,t7_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e5_7_1__calcul_2,t3_calcul_2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.65),file(calcul_2,e6_7_1__calcul_2),[file(calcul_2,e6_7_1__calcul_2)]]). fof(i3_7_1__calcul_2,theorem,( $true ), introduced(tautology,[file(calcul_2,i3_7_1__calcul_2)]), [interesting(0.65),trivial,file(calcul_2,i3_7_1__calcul_2)]). fof(i2_7_1__calcul_2,plain,( r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ), inference(conclusion,[status(thm),assumptions([dt_c1_7_1__calcul_2,e3_7_1__calcul_2,dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[e6_7_1__calcul_2,i3_7_1__calcul_2]), [interesting(0.65),file(calcul_2,i2_7_1__calcul_2),[file(calcul_2,i2_7_1__calcul_2)]]). fof(i1_7_1__calcul_2,plain,( ~ ( r2_hidden(c1_7_1__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) & ~ r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7_1__calcul_2,dt_c1_7__calcul_2,dt_c2_7__calcul_2]),discharge_asm(discharge,[e3_7_1__calcul_2])],[e3_7_1__calcul_2,i2_7_1__calcul_2]), [interesting(0.65),file(calcul_2,i1_7_1__calcul_2),[file(calcul_2,i1_7_1__calcul_2)]]). fof(i1_7_1_tmp__calcul_2,plain,( ~ ( r2_hidden(c1_7_1__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) & ~ r2_hidden(c1_7_1__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2]),discharge_asm(discharge,[dt_c1_7_1__calcul_2])],[dt_c1_7_1__calcul_2,i1_7_1__calcul_2]), [interesting(0.8),e1_7__calcul_2]). fof(e1_7__calcul_2,plain,( r1_tarski(k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1))),k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ), inference(let,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[i1_7_1_tmp__calcul_2,dt_k5_ordinal2,cc2_xreal_0,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_ordinal2,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,spc1_numerals,spc1_boole,d3_tarski,dh_c1_7_1__calcul_2]), [interesting(0.8),file(calcul_2,e1_7__calcul_2),[file(calcul_2,e1_7__calcul_2)]]). fof(dt_c3_7__calcul_2,assumption,( $true ), introduced(assumption,[file(calcul_2,c3_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c3_7__calcul_2)]). fof(dh_c3_7__calcul_2,definition, ( ~ ( r2_hidden(c3_7__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) & ~ r2_hidden(c3_7__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ) => ! [A] : ~ ( r2_hidden(A,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) & ~ r2_hidden(A,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ) ), introduced(definition,[new_symbol(c3_7__calcul_2),file(calcul_2,c3_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c3_7__calcul_2)]). fof(e2_7__calcul_2,assumption,( r2_hidden(c3_7__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) ), introduced(assumption,[file(calcul_2,e2_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,e2_7__calcul_2)]). fof(de_c4_7__calcul_2,definition,( c4_7__calcul_2 = c3_7__calcul_2 ), introduced(definition,[new_symbol(c4_7__calcul_2),file(calcul_2,c4_7__calcul_2)]), [interesting(0.8),axiom,file(calcul_2,c4_7__calcul_2)]). fof(e3_7__calcul_2,plain,( m2_subset_1(c3_7__calcul_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_tarski,dt_k1_xboole_0,cc2_xreal_0,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,fc2_finseq_1,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,dt_k1_calcul_2,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc1_ordinal2,fc3_nat_1,fc4_nat_1,spc6_arithm,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_calcul_2,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k2_calcul_2,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_calcul_2,dt_k2_xcmplx_0,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,t1_subset,t7_boole,spc1_numerals,spc1_boole,e2_7__calcul_2]), [interesting(0.8),file(calcul_2,e3_7__calcul_2),[file(calcul_2,e3_7__calcul_2)]]). fof(dt_c4_7__calcul_2,plain,( m2_subset_1(c4_7__calcul_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[cc1_finseq_1,cc2_funct_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc2_finseq_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc3_nat_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_funct_1,cc1_nat_1,cc2_int_1,cc2_nat_1,fc1_ordinal2,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c3_7__calcul_2,de_c4_7__calcul_2,e3_7__calcul_2]), [interesting(0.8),file(calcul_2,c4_7__calcul_2),[file(calcul_2,c4_7__calcul_2)]]). fof(t1_calcul_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ! [C] : ( v4_ordinal2(C) => ( r2_hidden(A,k2_calcul_2(B,C)) <=> ( r1_xreal_0(k2_xcmplx_0(1,B),A) & r1_xreal_0(A,k2_xcmplx_0(C,B)) ) ) ) ) ) ), file(calcul_2,t1_calcul_2), [interesting(0.9),axiom,file(calcul_2,t1_calcul_2)]). fof(e4_7__calcul_2,plain, ( r1_xreal_0(k2_xcmplx_0(1,c1_7__calcul_2),c4_7__calcul_2) & r1_xreal_0(c4_7__calcul_2,k2_xcmplx_0(k2_xcmplx_0(c2_7__calcul_2,1),c1_7__calcul_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,fc1_ordinal2,fc2_finseq_1,rc2_nat_1,rc3_nat_1,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,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,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_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,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,d1_calcul_2,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k2_calcul_2,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,dt_c4_7__calcul_2,de_c4_7__calcul_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,spc1_numerals,spc1_boole,e2_7__calcul_2,t1_calcul_2]), [interesting(0.8),file(calcul_2,e4_7__calcul_2),[file(calcul_2,e4_7__calcul_2)]]). fof(t26_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ~ ( r1_xreal_0(A,k2_xcmplx_0(B,1)) & ~ r1_xreal_0(A,B) & A != k2_xcmplx_0(B,1) ) ) ) ), file(nat_1,t26_nat_1), [interesting(0.9),axiom,file(nat_1,t26_nat_1)]). fof(e5_7__calcul_2,plain, ( r1_xreal_0(k2_xcmplx_0(1,c1_7__calcul_2),c4_7__calcul_2) & ( r1_xreal_0(c4_7__calcul_2,k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2)) | c4_7__calcul_2 = k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,fc1_ordinal2,fc2_finseq_1,rc2_nat_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,dt_c3_7__calcul_2,cc1_funct_1,cc1_nat_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,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,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_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,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c4_7__calcul_2,de_c4_7__calcul_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,spc1_numerals,spc1_boole,e4_7__calcul_2,t26_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(calcul_2,e5_7__calcul_2),[file(calcul_2,e5_7__calcul_2)]]). fof(e6_7__calcul_2,plain, ( r2_hidden(c4_7__calcul_2,k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2)) | r2_hidden(c4_7__calcul_2,k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,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_ordinal2,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc4_int_1,fc6_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_int_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c3_7__calcul_2,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_nat_1,fc1_xreal_0,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_nat_1,rc1_xreal_0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t3_subset,t4_arithm,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,d1_calcul_2,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c4_7__calcul_2,de_c4_7__calcul_2,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,t1_subset,t7_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e5_7__calcul_2,d1_tarski,rqLessOrEqual__r1_xreal_0__r1_r1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0]), [interesting(0.8),file(calcul_2,e6_7__calcul_2),[file(calcul_2,e6_7__calcul_2)]]). fof(e7_7__calcul_2,plain,( r2_hidden(c3_7__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[reflexivity_r1_xreal_0,connectedness_r1_xreal_0,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,cc2_xreal_0,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_ordinal2,fc2_finseq_1,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_boole,t2_tarski,fraenkel_a_2_0_calcul_2,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,spc6_arithm,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,d1_calcul_2,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,dt_c4_7__calcul_2,de_c4_7__calcul_2,t1_subset,t7_boole,spc1_numerals,spc1_boole,e6_7__calcul_2,d2_xboole_0]), [interesting(0.8),file(calcul_2,e7_7__calcul_2),[file(calcul_2,e7_7__calcul_2)]]). fof(i6_7__calcul_2,theorem,( $true ), introduced(tautology,[file(calcul_2,i6_7__calcul_2)]), [interesting(0.8),trivial,file(calcul_2,i6_7__calcul_2)]). fof(i5_7__calcul_2,plain,( r2_hidden(c3_7__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ), inference(conclusion,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2,e2_7__calcul_2])],[e7_7__calcul_2,i6_7__calcul_2]), [interesting(0.8),file(calcul_2,i5_7__calcul_2),[file(calcul_2,i5_7__calcul_2)]]). fof(i4_7__calcul_2,plain,( ~ ( r2_hidden(c3_7__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) & ~ r2_hidden(c3_7__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2,dt_c3_7__calcul_2]),discharge_asm(discharge,[e2_7__calcul_2])],[e2_7__calcul_2,i5_7__calcul_2]), [interesting(0.8),file(calcul_2,i4_7__calcul_2),[file(calcul_2,i4_7__calcul_2)]]). fof(i4_7_tmp__calcul_2,plain,( ~ ( r2_hidden(c3_7__calcul_2,k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1))) & ~ r2_hidden(c3_7__calcul_2,k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2]),discharge_asm(discharge,[dt_c3_7__calcul_2])],[dt_c3_7__calcul_2,i4_7__calcul_2]), [interesting(0.8),i3_7__calcul_2]). fof(i3_7__calcul_2,plain,( r1_tarski(k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1)),k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1)))) ), inference(let,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[i4_7_tmp__calcul_2,dt_k5_ordinal2,cc2_xreal_0,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_ordinal2,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,spc1_numerals,spc1_boole,d3_tarski,dh_c3_7__calcul_2]), [interesting(0.8),file(calcul_2,i3_7__calcul_2),[file(calcul_2,i3_7__calcul_2)]]). fof(i2_7__calcul_2,plain,( k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1)) ), inference(conclusion,[status(thm),assumptions([dt_c1_7__calcul_2,dt_c2_7__calcul_2])],[dt_k5_ordinal2,cc2_xreal_0,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_ordinal2,fc3_xreal_0,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_calcul_2,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_int_1,cc2_nat_1,cc3_int_1,cc3_nat_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,redefinition_k2_calcul_2,dt_k1_tarski,dt_k2_calcul_2,dt_k2_xboole_0,dt_k2_xcmplx_0,dt_c1_7__calcul_2,dt_c2_7__calcul_2,spc1_numerals,spc1_boole,d10_xboole_0,e1_7__calcul_2,i3_7__calcul_2]), [interesting(0.8),file(calcul_2,i2_7__calcul_2),[file(calcul_2,i2_7__calcul_2)]]). fof(i2_7_tmp__calcul_2,plain, ( v4_ordinal2(c2_7__calcul_2) => k2_xboole_0(k2_calcul_2(c1_7__calcul_2,c2_7__calcul_2),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,c2_7__calcul_2),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(c2_7__calcul_2,1)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__calcul_2]),discharge_asm(discharge,[dt_c2_7__calcul_2])],[dt_c2_7__calcul_2,i2_7__calcul_2]), [interesting(0.8),i1_7__calcul_2]). fof(i1_7__calcul_2,plain,( ! [A] : ( v4_ordinal2(A) => k2_xboole_0(k2_calcul_2(c1_7__calcul_2,A),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,A),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(A,1)) ) ), inference(let,[status(thm),assumptions([dt_c1_7__calcul_2])],[i2_7_tmp__calcul_2,dh_c2_7__calcul_2]), [interesting(0.8),file(calcul_2,i1_7__calcul_2),[file(calcul_2,i1_7__calcul_2)]]). fof(i1_7_tmp__calcul_2,plain, ( v4_ordinal2(c1_7__calcul_2) => ! [A] : ( v4_ordinal2(A) => k2_xboole_0(k2_calcul_2(c1_7__calcul_2,A),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(c1_7__calcul_2,A),1))) = k2_calcul_2(c1_7__calcul_2,k2_xcmplx_0(A,1)) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_7__calcul_2])],[dt_c1_7__calcul_2,i1_7__calcul_2]), [interesting(1),t5_calcul_2]). fof(t5_calcul_2,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => k2_xboole_0(k2_calcul_2(A,B),k1_tarski(k2_xcmplx_0(k2_xcmplx_0(A,B),1))) = k2_calcul_2(A,k2_xcmplx_0(B,1)) ) ) ), inference(let,[status(thm),assumptions([])],[i1_7_tmp__calcul_2,dh_c1_7__calcul_2]), [interesting(1),file(calcul_2,t5_calcul_2),[file(calcul_2,t5_calcul_2)]]).