% Mizar ND problem: t87_rfunct_3,rfunct_3,3337,34 fof(dh_c1_111__rfunct_3,definition, ( ( ~ v1_xboole_0(c1_111__rfunct_3) => ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_111__rfunct_3,k1_numbers) ) => ! [B,C] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,k2_xboole_0(B,C)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,B)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,C))) ) => k24_rfunct_3(c1_111__rfunct_3,A,k2_xboole_0(B,C)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,A,B),k24_rfunct_3(c1_111__rfunct_3,A,C)) ) ) ) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( ( v1_funct_1(E) & m2_relset_1(E,D,k1_numbers) ) => ! [F,G] : ( ( v1_finset_1(k4_relset_1(D,k1_numbers,k2_partfun1(D,k1_numbers,E,k2_xboole_0(F,G)))) & r1_xboole_0(k4_relset_1(D,k1_numbers,k2_partfun1(D,k1_numbers,E,F)),k4_relset_1(D,k1_numbers,k2_partfun1(D,k1_numbers,E,G))) ) => k24_rfunct_3(D,E,k2_xboole_0(F,G)) = k3_real_1(k24_rfunct_3(D,E,F),k24_rfunct_3(D,E,G)) ) ) ) ), introduced(definition,[new_symbol(c1_111__rfunct_3),file(rfunct_3,c1_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c1_111__rfunct_3)]). fof(dh_c2_111__rfunct_3,definition, ( ( ( v1_funct_1(c2_111__rfunct_3) & m2_relset_1(c2_111__rfunct_3,c1_111__rfunct_3,k1_numbers) ) => ! [A,B] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(A,B)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,A)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,B))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(A,B)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,A),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,B)) ) ) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,c1_111__rfunct_3,k1_numbers) ) => ! [D,E] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,C,k2_xboole_0(D,E)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,C,D)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,C,E))) ) => k24_rfunct_3(c1_111__rfunct_3,C,k2_xboole_0(D,E)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,C,D),k24_rfunct_3(c1_111__rfunct_3,C,E)) ) ) ), introduced(definition,[new_symbol(c2_111__rfunct_3),file(rfunct_3,c2_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c2_111__rfunct_3)]). fof(dh_c3_111__rfunct_3,definition, ( ! [A] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,A)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,A))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,A)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,A)) ) => ! [B,C] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(B,C)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,B)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,C))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(B,C)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,B),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,C)) ) ), introduced(definition,[new_symbol(c3_111__rfunct_3),file(rfunct_3,c3_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c3_111__rfunct_3)]). fof(dh_c4_111__rfunct_3,definition, ( ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3)) ) => ! [A] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,A)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,A))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,A)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,A)) ) ), introduced(definition,[new_symbol(c4_111__rfunct_3),file(rfunct_3,c4_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c4_111__rfunct_3)]). fof(e1_111__rfunct_3,assumption, ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ), introduced(assumption,[file(rfunct_3,e1_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,e1_111__rfunct_3)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(rc2_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc2_xreal_0)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_xreal_0,theorem,( ? [A] : ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ), file(xreal_0,rc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc4_xreal_0)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). 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_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_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(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_partfun1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_xboole_0(A) ) ), file(partfun1,rc1_partfun1), [interesting(0.9),axiom,file(partfun1,rc1_partfun1)]). fof(rc1_seq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_seq_1(A) ) ), file(seq_1,rc1_seq_1), [interesting(0.9),axiom,file(seq_1,rc1_seq_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_finset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), [interesting(0.9),axiom,file(finset_1,rc2_finset_1)]). fof(rc2_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_1)]). fof(rc3_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_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(rc4_finseq_1,theorem,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc4_finseq_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(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(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_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_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(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_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(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc1_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(nat_1,cc1_nat_1), [interesting(0.9),axiom,file(nat_1,cc1_nat_1)]). fof(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(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_finset_1,theorem,( ! [A] : ( v1_finset_1(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_finset_1(B) ) ) ), file(finset_1,cc2_finset_1), [interesting(0.9),axiom,file(finset_1,cc2_finset_1)]). fof(cc2_membered,theorem,( ! [A] : ( v4_membered(A) => v3_membered(A) ) ), file(membered,cc2_membered), [interesting(0.9),axiom,file(membered,cc2_membered)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc14_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_zfmisc_1(A,B)) ) ), file(finset_1,fc14_finset_1), [interesting(0.9),axiom,file(finset_1,fc14_finset_1)]). fof(fc22_membered,theorem,( ! [A,B] : ( ( v1_membered(A) & v1_membered(B) ) => v1_membered(k2_xboole_0(A,B)) ) ), file(membered,fc22_membered), [interesting(0.9),axiom,file(membered,fc22_membered)]). fof(fc23_membered,theorem,( ! [A,B] : ( ( v2_membered(A) & v2_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc23_membered), [interesting(0.9),axiom,file(membered,fc23_membered)]). fof(fc24_membered,theorem,( ! [A,B] : ( ( v3_membered(A) & v3_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc24_membered), [interesting(0.9),axiom,file(membered,fc24_membered)]). fof(fc25_membered,theorem,( ! [A,B] : ( ( v4_membered(A) & v4_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc25_membered), [interesting(0.9),axiom,file(membered,fc25_membered)]). fof(fc26_membered,theorem,( ! [A,B] : ( ( v5_membered(A) & v5_membered(B) ) => ( v1_membered(k2_xboole_0(A,B)) & v2_membered(k2_xboole_0(A,B)) & v3_membered(k2_xboole_0(A,B)) & v4_membered(k2_xboole_0(A,B)) & v5_membered(k2_xboole_0(A,B)) ) ) ), file(membered,fc26_membered), [interesting(0.9),axiom,file(membered,fc26_membered)]). 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(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(fc9_finset_1,theorem,( ! [A,B] : ( ( v1_finset_1(A) & v1_finset_1(B) ) => v1_finset_1(k2_xboole_0(A,B)) ) ), file(finset_1,fc9_finset_1), [interesting(0.9),axiom,file(finset_1,fc9_finset_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_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_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_rfinseq,theorem,( ? [A] : ( m1_finseq_1(A,k1_numbers) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) & v1_rfinseq(A) ) ), file(rfinseq,rc1_rfinseq), [interesting(0.9),axiom,file(rfinseq,rc1_rfinseq)]). fof(rc2_finseq_1,theorem,( ! [A] : ? [B] : ( m1_relset_1(B,k5_numbers,A) & v1_relat_1(B) & v1_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc2_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc2_finseq_1)]). fof(rc2_partfun1,theorem,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) ) ), file(partfun1,rc2_partfun1), [interesting(0.9),axiom,file(partfun1,rc2_partfun1)]). fof(rc2_rfunct_3,theorem,( ! [A,B] : ? [C] : ( m1_relset_1(C,A,B) & v1_relat_1(C) & v1_funct_1(C) & v1_finset_1(C) ) ), file(rfunct_3,rc2_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,rc2_rfunct_3)]). fof(rc3_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc3_finset_1), [interesting(0.9),axiom,file(finset_1,rc3_finset_1)]). fof(rc4_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) & v1_finset_1(B) ) ) ), file(finset_1,rc4_finset_1), [interesting(0.9),axiom,file(finset_1,rc4_finset_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(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(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(existence_m1_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(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_finseq_1,definition,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(dt_m1_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(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(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(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(cc1_finset_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_finset_1(A) ) ), file(finset_1,cc1_finset_1), [interesting(0.9),axiom,file(finset_1,cc1_finset_1)]). 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(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(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(dt_k15_rvsum_1,axiom,( ! [A] : ( m1_finseq_1(A,k1_numbers) => m1_subset_1(k15_rvsum_1(A),k1_numbers) ) ), file(rvsum_1,k15_rvsum_1), [interesting(0.9),axiom,file(rvsum_1,k15_rvsum_1)]). fof(dt_k22_rfunct_3,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) ) => ( v1_rfinseq(k22_rfunct_3(A,B,C)) & m2_finseq_1(k22_rfunct_3(A,B,C),k1_numbers) ) ) ), file(rfunct_3,k22_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,k22_rfunct_3)]). fof(dt_k24_rfunct_3,axiom,( ! [A,B,C] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) ) => m1_subset_1(k24_rfunct_3(A,B,C),k1_numbers) ) ), file(rfunct_3,k24_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,k24_rfunct_3)]). 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_c1_111__rfunct_3,assumption,( ~ v1_xboole_0(c1_111__rfunct_3) ), introduced(assumption,[file(rfunct_3,c1_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c1_111__rfunct_3)]). fof(dt_c2_111__rfunct_3,assumption, ( v1_funct_1(c2_111__rfunct_3) & m2_relset_1(c2_111__rfunct_3,c1_111__rfunct_3,k1_numbers) ), introduced(assumption,[file(rfunct_3,c2_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c2_111__rfunct_3)]). fof(dt_c3_111__rfunct_3,assumption,( $true ), introduced(assumption,[file(rfunct_3,c3_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c3_111__rfunct_3)]). fof(dt_c4_111__rfunct_3,assumption,( $true ), introduced(assumption,[file(rfunct_3,c4_111__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c4_111__rfunct_3)]). fof(d15_rfunct_3,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : k24_rfunct_3(A,B,C) = k15_rvsum_1(k22_rfunct_3(A,B,C)) ) ) ), file(rfunct_3,d15_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,d15_rfunct_3)]). fof(e1_111_3__rfunct_3,plain,( k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k5_ordinal2,cc1_xreal_0,cc2_xreal_0,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc5_membered,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,antisymmetry_r2_hidden,redefinition_k5_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_numbers,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc14_finset_1,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc6_membered,fc9_finset_1,rc1_finseq_1,rc1_finset_1,rc1_membered,rc1_rfinseq,rc2_finseq_1,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t1_boole,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k1_numbers,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,cc15_membered,cc1_finset_1,fc2_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,dt_k15_rvsum_1,dt_k22_rfunct_3,dt_k24_rfunct_3,dt_k2_xboole_0,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,d15_rfunct_3]), [interesting(0.65),file(rfunct_3,e1_111_3__rfunct_3),[file(rfunct_3,e1_111_3__rfunct_3)]]). fof(dt_k7_finseq_1,axiom,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,k7_finseq_1), [interesting(0.9),axiom,file(finseq_1,k7_finseq_1)]). fof(fc13_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(A,B)) & v1_relat_1(k7_finseq_1(A,B)) & v1_funct_1(k7_finseq_1(A,B)) & v1_finset_1(k7_finseq_1(A,B)) & v1_finseq_1(k7_finseq_1(A,B)) ) ) ), file(finseq_1,fc13_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc13_finseq_1)]). fof(fc14_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) & ~ v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) => ( ~ v1_xboole_0(k7_finseq_1(B,A)) & v1_relat_1(k7_finseq_1(B,A)) & v1_funct_1(k7_finseq_1(B,A)) & v1_finset_1(k7_finseq_1(B,A)) & v1_finseq_1(k7_finseq_1(B,A)) ) ) ), file(finseq_1,fc14_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc14_finseq_1)]). fof(symmetry_r1_rfinseq,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => ( r1_rfinseq(A,B) => r1_rfinseq(B,A) ) ) ), file(rfinseq,r1_rfinseq), [interesting(0.9),axiom,file(rfinseq,r1_rfinseq)]). fof(reflexivity_r1_rfinseq,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_relat_1(B) ) => r1_rfinseq(A,A) ) ), file(rfinseq,r1_rfinseq), [interesting(0.9),axiom,file(rfinseq,r1_rfinseq)]). fof(redefinition_k8_finseq_1,definition,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => k8_finseq_1(A,B,C) = k7_finseq_1(B,C) ) ), file(finseq_1,k8_finseq_1), [interesting(0.9),axiom,file(finseq_1,k8_finseq_1)]). fof(dt_k8_finseq_1,axiom,( ! [A,B,C] : ( ( m1_finseq_1(B,A) & m1_finseq_1(C,A) ) => m2_finseq_1(k8_finseq_1(A,B,C),A) ) ), file(finseq_1,k8_finseq_1), [interesting(0.9),axiom,file(finseq_1,k8_finseq_1)]). fof(fc2_seq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_seq_1(A) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_seq_1(k7_relat_1(A,B)) ) ) ), file(seq_1,fc2_seq_1), [interesting(0.9),axiom,file(seq_1,fc2_seq_1)]). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), [interesting(0.9),axiom,file(relat_1,k1_relat_1)]). fof(dt_k7_relat_1,axiom,( ! [A,B] : ( v1_relat_1(A) => v1_relat_1(k7_relat_1(A,B)) ) ), file(relat_1,k7_relat_1), [interesting(0.9),axiom,file(relat_1,k7_relat_1)]). fof(fc17_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k1_relat_1(A)) ) ), file(finseq_1,fc17_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc17_finseq_1)]). fof(fc1_rfinseq,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(B) ) => ( v1_relat_1(k7_relat_1(A,B)) & v1_funct_1(k7_relat_1(A,B)) & v1_finset_1(k7_relat_1(A,B)) ) ) ), file(rfinseq,fc1_rfinseq), [interesting(0.9),axiom,file(rfinseq,fc1_rfinseq)]). fof(commutativity_k4_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k4_subset_1(A,C,B) ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(idempotence_k4_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,B) = B ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(symmetry_r1_xboole_0,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), [interesting(0.9),axiom,file(xboole_0,r1_xboole_0)]). fof(redefinition_k2_partfun1,definition,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_partfun1(A,B,C,D) = k7_relat_1(C,D) ) ), file(partfun1,k2_partfun1), [interesting(0.9),axiom,file(partfun1,k2_partfun1)]). fof(redefinition_k4_relset_1,definition,( ! [A,B,C] : ( m1_relset_1(C,A,B) => k4_relset_1(A,B,C) = k1_relat_1(C) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(redefinition_k4_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k4_subset_1(A,B,C) = k2_xboole_0(B,C) ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(dt_k2_partfun1,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & m1_relset_1(C,A,B) ) => ( v1_funct_1(k2_partfun1(A,B,C,D)) & m2_relset_1(k2_partfun1(A,B,C,D),A,B) ) ) ), file(partfun1,k2_partfun1), [interesting(0.9),axiom,file(partfun1,k2_partfun1)]). fof(dt_k4_relset_1,axiom,( ! [A,B,C] : ( m1_relset_1(C,A,B) => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(relset_1,k4_relset_1), [interesting(0.9),axiom,file(relset_1,k4_relset_1)]). fof(dt_k4_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k4_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k4_subset_1), [interesting(0.9),axiom,file(subset_1,k4_subset_1)]). fof(fc10_finset_1,theorem,( ! [A,B] : ( v1_finset_1(B) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc10_finset_1), [interesting(0.9),axiom,file(finset_1,fc10_finset_1)]). fof(fc11_finset_1,theorem,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k3_xboole_0(A,B)) ) ), file(finset_1,fc11_finset_1), [interesting(0.9),axiom,file(finset_1,fc11_finset_1)]). 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(fc31_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc31_membered), [interesting(0.9),axiom,file(membered,fc31_membered)]). fof(fc32_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc32_membered), [interesting(0.9),axiom,file(membered,fc32_membered)]). fof(fc33_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc33_membered), [interesting(0.9),axiom,file(membered,fc33_membered)]). fof(fc34_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc34_membered), [interesting(0.9),axiom,file(membered,fc34_membered)]). fof(fc35_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) & v3_membered(k3_xboole_0(A,B)) & v4_membered(k3_xboole_0(A,B)) & v5_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc35_membered), [interesting(0.9),axiom,file(membered,fc35_membered)]). fof(fc36_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) & v3_membered(k3_xboole_0(B,A)) & v4_membered(k3_xboole_0(B,A)) & v5_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc36_membered), [interesting(0.9),axiom,file(membered,fc36_membered)]). fof(t2_boole,theorem,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), [interesting(0.9),axiom,file(boole,t2_boole)]). fof(fc27_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(A,B)) ) ), file(membered,fc27_membered), [interesting(0.9),axiom,file(membered,fc27_membered)]). fof(fc28_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k3_xboole_0(B,A)) ) ), file(membered,fc28_membered), [interesting(0.9),axiom,file(membered,fc28_membered)]). fof(fc29_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(A,B)) & v2_membered(k3_xboole_0(A,B)) ) ) ), file(membered,fc29_membered), [interesting(0.9),axiom,file(membered,fc29_membered)]). fof(fc30_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k3_xboole_0(B,A)) & v2_membered(k3_xboole_0(B,A)) ) ) ), file(membered,fc30_membered), [interesting(0.9),axiom,file(membered,fc30_membered)]). fof(commutativity_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(idempotence_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(commutativity_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(idempotence_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(redefinition_k5_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_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_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(dt_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(t68_funct_1,theorem,( ! [A,B] : ( ( v1_relat_1(B) & v1_funct_1(B) ) => ! [C] : ( ( v1_relat_1(C) & v1_funct_1(C) ) => ( B = k7_relat_1(C,A) <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A) & ! [D] : ( r2_hidden(D,k1_relat_1(B)) => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ), file(funct_1,t68_funct_1), [interesting(0.9),axiom,file(funct_1,t68_funct_1)]). fof(e1_111_2__rfunct_3,plain,( k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))))) = k5_subset_1(c1_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc1_seq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t2_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,antisymmetry_r2_hidden,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k5_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k5_subset_1,dt_k7_relat_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t1_subset,t7_boole,t68_funct_1]), [interesting(0.65),file(rfunct_3,e1_111_2__rfunct_3),[file(rfunct_3,e1_111_2__rfunct_3)]]). fof(e2_111_2__rfunct_3,plain,( k5_subset_1(c1_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) = k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc1_seq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t2_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,antisymmetry_r2_hidden,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k5_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k5_subset_1,dt_k7_relat_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t1_subset,t7_boole,t68_funct_1]), [interesting(0.65),file(rfunct_3,e2_111_2__rfunct_3),[file(rfunct_3,e2_111_2__rfunct_3)]]). fof(t16_xboole_1,theorem,( ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ), file(xboole_1,t16_xboole_1), [interesting(0.9),axiom,file(xboole_1,t16_xboole_1)]). fof(e3_111_2__rfunct_3,plain,( k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) = k3_xboole_0(k5_subset_1(c1_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3)),k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_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_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t1_subset,t2_boole,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,redefinition_k4_relset_1,redefinition_k5_subset_1,dt_k1_numbers,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k5_subset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t16_xboole_1]), [interesting(0.65),file(rfunct_3,e3_111_2__rfunct_3),[file(rfunct_3,e3_111_2__rfunct_3)]]). fof(e4_111_2__rfunct_3,plain,( k3_xboole_0(k5_subset_1(c1_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3)),k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc1_seq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t2_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,antisymmetry_r2_hidden,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k5_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k5_subset_1,dt_k7_relat_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t1_subset,t7_boole,t68_funct_1]), [interesting(0.65),file(rfunct_3,e4_111_2__rfunct_3),[file(rfunct_3,e4_111_2__rfunct_3)]]). fof(e6_111__rfunct_3,plain,( k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))))) = k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[e1_111_2__rfunct_3,e2_111_2__rfunct_3,e3_111_2__rfunct_3,e4_111_2__rfunct_3]), [interesting(0.8),file(rfunct_3,e6_111__rfunct_3),[file(rfunct_3,e6_111__rfunct_3)]]). fof(e1_111_1__rfunct_3,plain,( k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) = k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc1_seq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t2_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,redefinition_k2_partfun1,redefinition_k4_relset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k7_relat_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t1_subset,t7_boole,t68_funct_1]), [interesting(0.65),file(rfunct_3,e1_111_1__rfunct_3),[file(rfunct_3,e1_111_1__rfunct_3)]]). fof(t23_xboole_1,theorem,( ! [A,B,C] : k3_xboole_0(A,k2_xboole_0(B,C)) = k2_xboole_0(k3_xboole_0(A,B),k3_xboole_0(A,C)) ), file(xboole_1,t23_xboole_1), [interesting(0.9),axiom,file(xboole_1,t23_xboole_1)]). fof(e2_111_1__rfunct_3,plain,( k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k2_xboole_0(k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),c3_111__rfunct_3),k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),c4_111__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_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_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t1_subset,t2_boole,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,redefinition_k4_relset_1,dt_k1_numbers,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t23_xboole_1]), [interesting(0.65),file(rfunct_3,e2_111_1__rfunct_3),[file(rfunct_3,e2_111_1__rfunct_3)]]). fof(e3_111_1__rfunct_3,plain,( k2_xboole_0(k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),c3_111__rfunct_3),k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),c4_111__rfunct_3)) = k2_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),c4_111__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc1_seq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t2_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,redefinition_k2_partfun1,redefinition_k4_relset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k7_relat_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t1_subset,t7_boole,t68_funct_1]), [interesting(0.65),file(rfunct_3,e3_111_1__rfunct_3),[file(rfunct_3,e3_111_1__rfunct_3)]]). fof(e4_111_1__rfunct_3,plain,( k2_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k3_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3),c4_111__rfunct_3)) = k4_subset_1(c1_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc1_seq_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t2_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,rc2_partfun1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,antisymmetry_r2_hidden,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k4_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k4_subset_1,dt_k7_relat_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t1_subset,t7_boole,t68_funct_1]), [interesting(0.65),file(rfunct_3,e4_111_1__rfunct_3),[file(rfunct_3,e4_111_1__rfunct_3)]]). fof(e2_111__rfunct_3,plain,( k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) = k4_subset_1(c1_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_nat_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,dt_k2_zfmisc_1,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc10_finset_1,fc11_finset_1,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc24_membered,fc25_membered,fc26_membered,fc2_seq_1,fc31_membered,fc32_membered,fc33_membered,fc34_membered,fc35_membered,fc36_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k7_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,fc27_membered,fc28_membered,fc29_membered,fc30_membered,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k4_subset_1,dt_k1_numbers,dt_k2_partfun1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_relset_1,dt_k4_subset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,e1_111_1__rfunct_3,e2_111_1__rfunct_3,e3_111_1__rfunct_3,e4_111_1__rfunct_3]), [interesting(0.8),file(rfunct_3,e2_111__rfunct_3),[file(rfunct_3,e2_111__rfunct_3)]]). fof(t7_xboole_1,theorem,( ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ), file(xboole_1,t7_xboole_1), [interesting(0.9),axiom,file(xboole_1,t7_xboole_1)]). fof(e3_111__rfunct_3,plain, ( r1_tarski(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) & r1_tarski(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,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_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc7_xreal_0,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k7_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,reflexivity_r1_tarski,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k4_subset_1,dt_k1_numbers,dt_k2_partfun1,dt_k2_xboole_0,dt_k4_relset_1,dt_k4_subset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t3_subset,e2_111__rfunct_3,t7_xboole_1]), [interesting(0.8),file(rfunct_3,e3_111__rfunct_3),[file(rfunct_3,e3_111__rfunct_3)]]). fof(t13_finset_1,theorem,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), [interesting(0.9),axiom,file(finset_1,t13_finset_1)]). fof(e4_111__rfunct_3,plain, ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3))) & v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,e1_111__rfunct_3])],[cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,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_xreal_0,cc3_membered,cc7_xreal_0,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc2_seq_1,fc6_membered,rc1_membered,rc1_seq_1,rc1_xreal_0,rc2_partfun1,rc2_rfunct_3,rc7_finseq_1,t1_boole,t1_subset,t4_subset,t5_subset,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k7_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_seq_1,cc2_finset_1,cc4_membered,cc6_membered,fc22_membered,fc23_membered,rc1_finset_1,rc3_finset_1,rc4_finset_1,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,symmetry_r1_xboole_0,redefinition_k2_partfun1,redefinition_k4_relset_1,dt_k1_numbers,dt_k2_partfun1,dt_k2_xboole_0,dt_k4_relset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,fc9_finset_1,t3_subset,e3_111__rfunct_3,e1_111__rfunct_3,t13_finset_1]), [interesting(0.8),file(rfunct_3,e4_111__rfunct_3),[file(rfunct_3,e4_111__rfunct_3)]]). fof(t66_rfunct_3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C))) => k22_rfunct_3(A,B,k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C))) = k22_rfunct_3(A,B,C) ) ) ) ), file(rfunct_3,t66_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,t66_rfunct_3)]). fof(e5_111__rfunct_3,plain, ( k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) & k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3) = k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3))) & k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3) = k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,e1_111__rfunct_3])],[dt_k5_ordinal2,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc5_membered,rc2_finset_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,existence_m1_finseq_1,redefinition_k5_numbers,dt_k5_numbers,dt_m1_finseq_1,cc1_finseq_1,cc1_nat_1,cc1_xreal_0,cc2_nat_1,cc2_xreal_0,cc3_nat_1,cc7_xreal_0,cc9_membered,fc17_finseq_1,fc1_rfinseq,fc2_seq_1,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_rfinseq,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_partfun1,rc2_rfunct_3,rc3_finseq_1,rc4_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k7_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc6_membered,fc14_finset_1,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc6_membered,rc1_membered,rc3_finset_1,rc4_finset_1,t1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,symmetry_r1_xboole_0,existence_m2_relset_1,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k22_rfunct_3,dt_k2_partfun1,dt_k2_xboole_0,dt_k4_relset_1,dt_m2_relset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,fc9_finset_1,rc1_finset_1,t6_boole,t7_boole,t8_boole,e4_111__rfunct_3,e1_111__rfunct_3,t66_rfunct_3]), [interesting(0.8),file(rfunct_3,e5_111__rfunct_3),[file(rfunct_3,e5_111__rfunct_3)]]). fof(t79_rfunct_3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C,D] : ( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_xboole_0(C,D)))) & r1_xboole_0(C,D) ) => r1_rfinseq(k22_rfunct_3(A,B,k2_xboole_0(C,D)),k8_finseq_1(k1_numbers,k22_rfunct_3(A,B,C),k22_rfunct_3(A,B,D))) ) ) ) ), file(rfunct_3,t79_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,t79_rfunct_3)]). fof(e7_111__rfunct_3,plain,( r1_rfinseq(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)),k8_finseq_1(k1_numbers,k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,e1_111__rfunct_3])],[dt_k5_ordinal2,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,fc5_membered,rc2_finset_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,redefinition_k5_numbers,dt_k5_numbers,cc1_finseq_1,cc1_nat_1,cc1_xreal_0,cc2_nat_1,cc2_xreal_0,cc3_nat_1,cc7_xreal_0,cc9_membered,fc13_finseq_1,fc14_finseq_1,fc2_seq_1,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_rfinseq,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc3_finseq_1,rc4_finseq_1,rc6_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k7_finseq_1,dt_k7_relat_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc4_membered,cc6_membered,fc14_finset_1,fc17_finseq_1,fc1_rfinseq,fc22_membered,fc23_membered,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc6_membered,rc1_membered,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k4_subset_1,idempotence_k4_subset_1,symmetry_r1_rfinseq,reflexivity_r1_rfinseq,symmetry_r1_xboole_0,existence_m2_relset_1,redefinition_k2_partfun1,redefinition_k4_relset_1,redefinition_k4_subset_1,redefinition_k8_finseq_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k22_rfunct_3,dt_k2_partfun1,dt_k2_xboole_0,dt_k4_relset_1,dt_k4_subset_1,dt_k8_finseq_1,dt_m2_relset_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,fc9_finset_1,rc1_finset_1,t6_boole,t7_boole,t8_boole,e6_111__rfunct_3,e1_111__rfunct_3,e2_111__rfunct_3,e5_111__rfunct_3,t79_rfunct_3]), [interesting(0.8),file(rfunct_3,e7_111__rfunct_3),[file(rfunct_3,e7_111__rfunct_3)]]). fof(t22_rfinseq,theorem,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => ( r1_rfinseq(A,B) => k15_rvsum_1(A) = k15_rvsum_1(B) ) ) ) ), file(rfinseq,t22_rfinseq), [interesting(0.9),axiom,file(rfinseq,t22_rfinseq)]). fof(e2_111_3__rfunct_3,plain,( k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3))) = k15_rvsum_1(k8_finseq_1(k1_numbers,k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,e1_111__rfunct_3])],[reflexivity_r1_tarski,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_partfun1,rc2_finset_1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_finseq_1,rc4_xreal_0,rc6_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc14_finset_1,fc24_membered,fc25_membered,fc26_membered,fc2_finseq_1,fc5_membered,fc6_membered,fc9_finset_1,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_rfinseq,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_nat_1,rc2_rfunct_3,rc3_finset_1,rc3_nat_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t1_boole,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k5_numbers,dt_k7_finseq_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc1_finseq_1,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc4_membered,fc13_finseq_1,fc14_finseq_1,fc22_membered,fc23_membered,rc1_finseq_1,rc2_partfun1,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,symmetry_r1_rfinseq,reflexivity_r1_rfinseq,existence_m2_finseq_1,redefinition_k8_finseq_1,redefinition_m2_finseq_1,dt_k15_rvsum_1,dt_k1_numbers,dt_k22_rfunct_3,dt_k2_xboole_0,dt_k8_finseq_1,dt_m2_finseq_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,e7_111__rfunct_3,t22_rfinseq]), [interesting(0.65),file(rfunct_3,e2_111_3__rfunct_3),[file(rfunct_3,e2_111_3__rfunct_3)]]). 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(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(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(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(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_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(commutativity_k3_real_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k3_real_1(B,A) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(commutativity_k9_binop_2,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k9_binop_2(B,A) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(redefinition_k3_real_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(redefinition_k9_binop_2,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k9_binop_2(A,B) = k2_xcmplx_0(A,B) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(dt_k3_real_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k3_real_1(A,B),k1_numbers) ) ), file(real_1,k3_real_1), [interesting(0.9),axiom,file(real_1,k3_real_1)]). fof(dt_k9_binop_2,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k9_binop_2(A,B),k1_numbers) ) ), file(binop_2,k9_binop_2), [interesting(0.9),axiom,file(binop_2,k9_binop_2)]). fof(t105_rvsum_1,theorem,( ! [A] : ( m2_finseq_1(A,k1_numbers) => ! [B] : ( m2_finseq_1(B,k1_numbers) => k15_rvsum_1(k8_finseq_1(k1_numbers,A,B)) = k9_binop_2(k15_rvsum_1(A),k15_rvsum_1(B)) ) ) ), file(rvsum_1,t105_rvsum_1), [interesting(0.9),axiom,file(rvsum_1,t105_rvsum_1)]). fof(e3_111_3__rfunct_3,plain,( k15_rvsum_1(k8_finseq_1(k1_numbers,k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) = k3_real_1(k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3)),k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[reflexivity_r1_tarski,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_partfun1,rc2_finset_1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_finseq_1,rc4_xreal_0,rc6_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc7_xreal_0,cc9_membered,fc13_finseq_1,fc14_finseq_1,fc14_finset_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_membered,fc6_membered,fc7_xreal_0,rc1_finseq_1,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_rfinseq,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_nat_1,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc3_nat_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_k5_numbers,redefinition_m2_relset_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_k7_finseq_1,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc4_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,commutativity_k9_binop_2,existence_m2_finseq_1,redefinition_k3_real_1,redefinition_k8_finseq_1,redefinition_k9_binop_2,redefinition_m2_finseq_1,dt_k15_rvsum_1,dt_k1_numbers,dt_k22_rfunct_3,dt_k3_real_1,dt_k8_finseq_1,dt_k9_binop_2,dt_m2_finseq_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,fc2_membered,t105_rvsum_1]), [interesting(0.65),file(rfunct_3,e3_111_3__rfunct_3),[file(rfunct_3,e3_111_3__rfunct_3)]]). fof(e4_111_3__rfunct_3,plain,( k3_real_1(k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3)),k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc8_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k5_ordinal2,cc1_xreal_0,cc2_xreal_0,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_nat_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_membered,fc7_xreal_0,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,antisymmetry_r2_hidden,redefinition_k5_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_numbers,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc14_finset_1,fc2_finseq_1,fc6_membered,rc1_finseq_1,rc1_finset_1,rc1_membered,rc1_rfinseq,rc2_finseq_1,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,cc15_membered,cc1_finset_1,fc2_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,redefinition_k3_real_1,dt_k15_rvsum_1,dt_k22_rfunct_3,dt_k24_rfunct_3,dt_k3_real_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,d15_rfunct_3]), [interesting(0.65),file(rfunct_3,e4_111_3__rfunct_3),[file(rfunct_3,e4_111_3__rfunct_3)]]). fof(e5_111_3__rfunct_3,plain,( k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k15_rvsum_1(k22_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3))) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc8_xreal_0,fc9_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,reflexivity_r1_tarski,dt_k5_ordinal2,cc1_xreal_0,cc2_xreal_0,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_nat_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc5_membered,fc7_xreal_0,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,antisymmetry_r2_hidden,redefinition_k5_numbers,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_k5_numbers,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc3_membered,cc4_membered,cc6_membered,cc9_membered,fc14_finset_1,fc2_finseq_1,fc6_membered,rc1_finseq_1,rc1_finset_1,rc1_membered,rc1_rfinseq,rc2_finseq_1,rc2_partfun1,rc2_rfunct_3,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_relset_1,redefinition_m2_finseq_1,redefinition_m2_relset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_relset_1,cc15_membered,cc1_finset_1,fc2_membered,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k3_real_1,redefinition_k3_real_1,dt_k15_rvsum_1,dt_k22_rfunct_3,dt_k24_rfunct_3,dt_k3_real_1,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,d15_rfunct_3]), [interesting(0.65),file(rfunct_3,e5_111_3__rfunct_3),[file(rfunct_3,e5_111_3__rfunct_3)]]). fof(e8_111__rfunct_3,plain,( k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3)) ), inference(iterative_eq,[status(thm),assumptions([e1_111__rfunct_3,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[e1_111_3__rfunct_3,e2_111_3__rfunct_3,e3_111_3__rfunct_3,e4_111_3__rfunct_3,e5_111_3__rfunct_3]), [interesting(0.8),file(rfunct_3,e8_111__rfunct_3),[file(rfunct_3,e8_111__rfunct_3)]]). fof(i5_111__rfunct_3,theorem,( $true ), introduced(tautology,[file(rfunct_3,i5_111__rfunct_3)]), [interesting(0.8),trivial,file(rfunct_3,i5_111__rfunct_3)]). fof(i4_111__rfunct_3,plain,( k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3)) ), inference(conclusion,[status(thm),assumptions([e1_111__rfunct_3,dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[e8_111__rfunct_3,i5_111__rfunct_3]), [interesting(0.8),file(rfunct_3,i4_111__rfunct_3),[file(rfunct_3,i4_111__rfunct_3)]]). fof(i3_111__rfunct_3,plain, ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3,dt_c3_111__rfunct_3,dt_c4_111__rfunct_3]),discharge_asm(discharge,[e1_111__rfunct_3])],[e1_111__rfunct_3,i4_111__rfunct_3]), [interesting(0.8),file(rfunct_3,i3_111__rfunct_3),[file(rfunct_3,i3_111__rfunct_3)]]). fof(i3_111_tmp__rfunct_3,plain, ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c3_111__rfunct_3)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,c4_111__rfunct_3))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(c3_111__rfunct_3,c4_111__rfunct_3)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c3_111__rfunct_3),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,c4_111__rfunct_3)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3]),discharge_asm(discharge,[dt_c3_111__rfunct_3,dt_c4_111__rfunct_3])],[dt_c3_111__rfunct_3,dt_c4_111__rfunct_3,i3_111__rfunct_3]), [interesting(0.8),i2_111__rfunct_3]). fof(i2_111__rfunct_3,plain,( ! [A,B] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(A,B)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,A)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,B))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(A,B)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,A),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,B)) ) ), inference(let,[status(thm),assumptions([dt_c1_111__rfunct_3,dt_c2_111__rfunct_3])],[i3_111_tmp__rfunct_3,dh_c3_111__rfunct_3,dh_c4_111__rfunct_3]), [interesting(0.8),file(rfunct_3,i2_111__rfunct_3),[file(rfunct_3,i2_111__rfunct_3)]]). fof(i2_111_tmp__rfunct_3,plain, ( ( v1_funct_1(c2_111__rfunct_3) & m2_relset_1(c2_111__rfunct_3,c1_111__rfunct_3,k1_numbers) ) => ! [A,B] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,k2_xboole_0(A,B)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,A)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,c2_111__rfunct_3,B))) ) => k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,k2_xboole_0(A,B)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,A),k24_rfunct_3(c1_111__rfunct_3,c2_111__rfunct_3,B)) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_111__rfunct_3]),discharge_asm(discharge,[dt_c2_111__rfunct_3])],[dt_c2_111__rfunct_3,i2_111__rfunct_3]), [interesting(0.8),i1_111__rfunct_3]). fof(i1_111__rfunct_3,plain,( ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_111__rfunct_3,k1_numbers) ) => ! [B,C] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,k2_xboole_0(B,C)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,B)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,C))) ) => k24_rfunct_3(c1_111__rfunct_3,A,k2_xboole_0(B,C)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,A,B),k24_rfunct_3(c1_111__rfunct_3,A,C)) ) ) ), inference(let,[status(thm),assumptions([dt_c1_111__rfunct_3])],[i2_111_tmp__rfunct_3,dh_c2_111__rfunct_3]), [interesting(0.8),file(rfunct_3,i1_111__rfunct_3),[file(rfunct_3,i1_111__rfunct_3)]]). fof(i1_111_tmp__rfunct_3,plain, ( ~ v1_xboole_0(c1_111__rfunct_3) => ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_111__rfunct_3,k1_numbers) ) => ! [B,C] : ( ( v1_finset_1(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,k2_xboole_0(B,C)))) & r1_xboole_0(k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,B)),k4_relset_1(c1_111__rfunct_3,k1_numbers,k2_partfun1(c1_111__rfunct_3,k1_numbers,A,C))) ) => k24_rfunct_3(c1_111__rfunct_3,A,k2_xboole_0(B,C)) = k3_real_1(k24_rfunct_3(c1_111__rfunct_3,A,B),k24_rfunct_3(c1_111__rfunct_3,A,C)) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_111__rfunct_3])],[dt_c1_111__rfunct_3,i1_111__rfunct_3]), [interesting(1),t87_rfunct_3]). fof(t87_rfunct_3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C,D] : ( ( v1_finset_1(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,k2_xboole_0(C,D)))) & r1_xboole_0(k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,C)),k4_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,B,D))) ) => k24_rfunct_3(A,B,k2_xboole_0(C,D)) = k3_real_1(k24_rfunct_3(A,B,C),k24_rfunct_3(A,B,D)) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_111_tmp__rfunct_3,dh_c1_111__rfunct_3]), [interesting(1),file(rfunct_3,t87_rfunct_3),[file(rfunct_3,t87_rfunct_3)]]).