% Mizar ND problem: t39_rfunct_3,rfunct_3,1315,51 fof(dh_c1_56__rfunct_3,definition, ( ( ~ v1_xboole_0(c1_56__rfunct_3) => ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_56__rfunct_3,k1_numbers) ) => k11_relset_1(c1_56__rfunct_3,k1_numbers,A,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,A),k2_setwiseo(k1_numbers,0)) ) ) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,B,k1_numbers) ) => k11_relset_1(B,k1_numbers,C,k2_limfunc1(0)) = k11_relset_1(B,k1_numbers,k19_rfunct_3(B,C),k2_setwiseo(k1_numbers,0)) ) ) ), introduced(definition,[new_symbol(c1_56__rfunct_3),file(rfunct_3,c1_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c1_56__rfunct_3)]). fof(dh_c2_56__rfunct_3,definition, ( ( ( v1_funct_1(c2_56__rfunct_3) & m2_relset_1(c2_56__rfunct_3,c1_56__rfunct_3,k1_numbers) ) => k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0)) ) => ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_56__rfunct_3,k1_numbers) ) => k11_relset_1(c1_56__rfunct_3,k1_numbers,A,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,A),k2_setwiseo(k1_numbers,0)) ) ), introduced(definition,[new_symbol(c2_56__rfunct_3),file(rfunct_3,c2_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c2_56__rfunct_3)]). fof(fc10_membered,theorem,( ! [A] : ( v1_int_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) ) ) ), file(membered,fc10_membered), [interesting(0.9),axiom,file(membered,fc10_membered)]). fof(fc9_membered,theorem,( ! [A] : ( v1_rat_1(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) ) ) ), file(membered,fc9_membered), [interesting(0.9),axiom,file(membered,fc9_membered)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_membered,theorem,( ! [A] : ( v5_membered(A) => v4_membered(A) ) ), file(membered,cc1_membered), [interesting(0.9),axiom,file(membered,cc1_membered)]). fof(cc1_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_xreal_0,theorem,( ! [A] : ( v4_ordinal2(A) => v1_xreal_0(A) ) ), file(xreal_0,cc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc1_xreal_0)]). fof(cc20_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) & v5_membered(B) ) ) ) ), file(membered,cc20_membered), [interesting(0.9),axiom,file(membered,cc20_membered)]). fof(cc2_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(cc3_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(fc11_membered,theorem,( ! [A] : ( v4_ordinal2(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) & v3_membered(k1_tarski(A)) & v4_membered(k1_tarski(A)) & v5_membered(k1_tarski(A)) ) ) ), file(membered,fc11_membered), [interesting(0.9),axiom,file(membered,fc11_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(fc18_finseq_1,theorem,( ! [A,B] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) => v1_finset_1(k10_relat_1(A,B)) ) ), file(finseq_1,fc18_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc18_finseq_1)]). 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(fc7_membered,theorem,( ! [A] : ( v1_xcmplx_0(A) => v1_membered(k1_tarski(A)) ) ), file(membered,fc7_membered), [interesting(0.9),axiom,file(membered,fc7_membered)]). 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_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_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_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_1)]). fof(rc2_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_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_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(rc3_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), [interesting(0.9),axiom,file(nat_1,rc3_nat_1)]). fof(rc3_xreal_0,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & v3_xreal_0(A) ) ), file(xreal_0,rc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc3_xreal_0)]). fof(rc4_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(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(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(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k10_relat_1,axiom,( $true ), file(relat_1,k10_relat_1), [interesting(0.9),axiom,file(relat_1,k10_relat_1)]). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k5_finsub_1,axiom,( ! [A] : v4_finsub_1(k5_finsub_1(A)) ), file(finsub_1,k5_finsub_1), [interesting(0.9),axiom,file(finsub_1,k5_finsub_1)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_m1_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_relset_1,axiom,( ! [A,B,C] : ( m2_relset_1(C,A,B) => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc1_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(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_seq_1,theorem,( ! [A,B] : ( v2_membered(B) => ! [C] : ( m1_relset_1(C,A,B) => ( v1_funct_1(C) => ( v1_funct_1(C) & v1_seq_1(C) ) ) ) ) ), file(seq_1,cc1_seq_1), [interesting(0.9),axiom,file(seq_1,cc1_seq_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(cc2_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => v1_xcmplx_0(A) ) ), file(xreal_0,cc2_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]). fof(cc3_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_membered,theorem,( ! [A] : ( v2_membered(A) => v1_membered(A) ) ), file(membered,cc4_membered), [interesting(0.9),axiom,file(membered,cc4_membered)]). fof(cc6_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) => ( v1_membered(A) & v2_membered(A) ) ) ), file(membered,cc6_membered), [interesting(0.9),axiom,file(membered,cc6_membered)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(cc9_membered,theorem,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(k5_numbers)) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc9_membered), [interesting(0.9),axiom,file(membered,cc9_membered)]). fof(fc1_finset_1,theorem,( ! [A] : ( ~ v1_xboole_0(k1_tarski(A)) & v1_finset_1(k1_tarski(A)) ) ), file(finset_1,fc1_finset_1), [interesting(0.9),axiom,file(finset_1,fc1_finset_1)]). fof(fc8_membered,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_membered(k1_tarski(A)) & v2_membered(k1_tarski(A)) ) ) ), file(membered,fc8_membered), [interesting(0.9),axiom,file(membered,fc8_membered)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(redefinition_k11_relset_1,definition,( ! [A,B,C,D] : ( m1_relset_1(C,A,B) => k11_relset_1(A,B,C,D) = k10_relat_1(C,D) ) ), file(relset_1,k11_relset_1), [interesting(0.9),axiom,file(relset_1,k11_relset_1)]). fof(redefinition_k2_setwiseo,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k2_setwiseo(A,B) = k1_tarski(B) ) ), file(setwiseo,k2_setwiseo), [interesting(0.9),axiom,file(setwiseo,k2_setwiseo)]). fof(dt_k11_relset_1,axiom,( ! [A,B,C,D] : ( m1_relset_1(C,A,B) => m1_subset_1(k11_relset_1(A,B,C,D),k1_zfmisc_1(A)) ) ), file(relset_1,k11_relset_1), [interesting(0.9),axiom,file(relset_1,k11_relset_1)]). fof(dt_k19_rfunct_3,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & v1_funct_1(B) & m1_relset_1(B,A,k1_numbers) ) => ( v1_funct_1(k19_rfunct_3(A,B)) & m2_relset_1(k19_rfunct_3(A,B),A,k1_numbers) ) ) ), file(rfunct_3,k19_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,k19_rfunct_3)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k2_limfunc1,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k2_limfunc1(A),k1_zfmisc_1(k1_numbers)) ) ), file(limfunc1,k2_limfunc1), [interesting(0.9),axiom,file(limfunc1,k2_limfunc1)]). fof(dt_k2_setwiseo,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k2_setwiseo(A,B),k5_finsub_1(A)) ) ), file(setwiseo,k2_setwiseo), [interesting(0.9),axiom,file(setwiseo,k2_setwiseo)]). fof(dt_c1_56__rfunct_3,assumption,( ~ v1_xboole_0(c1_56__rfunct_3) ), introduced(assumption,[file(rfunct_3,c1_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c1_56__rfunct_3)]). fof(dt_c2_56__rfunct_3,assumption, ( v1_funct_1(c2_56__rfunct_3) & m2_relset_1(c2_56__rfunct_3,c1_56__rfunct_3,k1_numbers) ), introduced(assumption,[file(rfunct_3,c2_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c2_56__rfunct_3)]). fof(fc2_membered,theorem, ( ~ v1_xboole_0(k1_numbers) & v1_membered(k1_numbers) & v2_membered(k1_numbers) ), file(membered,fc2_membered), [interesting(0.9),axiom,file(membered,fc2_membered)]). fof(spc0_numerals,theorem, ( v2_xreal_0(0) & m2_subset_1(0,k1_numbers,k5_numbers) & m1_subset_1(0,k5_numbers) & m1_subset_1(0,k1_numbers) ), file(numerals,spc0_numerals), [interesting(0.9),axiom,file(numerals,spc0_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(d10_xboole_0,definition,( ! [A,B] : ( A = B <=> ( r1_tarski(A,B) & r1_tarski(B,A) ) ) ), file(xboole_0,d10_xboole_0), [interesting(0.9),axiom,file(xboole_0,d10_xboole_0)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_c1_56_1__rfunct_3,assumption,( $true ), introduced(assumption,[file(rfunct_3,c1_56_1__rfunct_3)]), [interesting(0.65),axiom,file(rfunct_3,c1_56_1__rfunct_3)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(dh_c1_56_1__rfunct_3,definition, ( ~ ( r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) & ~ r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ) => ! [A] : ~ ( r2_hidden(A,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) & ~ r2_hidden(A,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ) ), introduced(definition,[new_symbol(c1_56_1__rfunct_3),file(rfunct_3,c1_56_1__rfunct_3)]), [interesting(0.65),axiom,file(rfunct_3,c1_56_1__rfunct_3)]). fof(e1_56_1__rfunct_3,assumption,( r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ), introduced(assumption,[file(rfunct_3,e1_56_1__rfunct_3)]), [interesting(0.65),axiom,file(rfunct_3,e1_56_1__rfunct_3)]). 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(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_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(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_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(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(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(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(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). 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_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(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(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_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(redefinition_k2_seq_1,definition,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => k2_seq_1(A,B,C,D) = k1_funct_1(C,D) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(redefinition_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(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(dt_k1_relat_1,axiom,( $true ), file(relat_1,k1_relat_1), [interesting(0.9),axiom,file(relat_1,k1_relat_1)]). fof(dt_k2_seq_1,axiom,( ! [A,B,C,D] : ( ( v2_membered(B) & v1_funct_1(C) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_seq_1(A,B,C,D),k1_numbers) ) ), file(seq_1,k2_seq_1), [interesting(0.9),axiom,file(seq_1,k2_seq_1)]). fof(dt_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(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(de_c2_56_1__rfunct_3,definition,( c2_56_1__rfunct_3 = c1_56_1__rfunct_3 ), introduced(definition,[new_symbol(c2_56_1__rfunct_3),file(rfunct_3,c2_56_1__rfunct_3)]), [interesting(0.65),axiom,file(rfunct_3,c2_56_1__rfunct_3)]). fof(e2_56_1__rfunct_3,plain,( m1_subset_1(c1_56_1__rfunct_3,c1_56__rfunct_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc18_finseq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_relat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k11_relset_1,dt_k11_relset_1,dt_k1_numbers,dt_k2_limfunc1,dt_m1_subset_1,dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e1_56_1__rfunct_3]), [interesting(0.65),file(rfunct_3,e2_56_1__rfunct_3),[file(rfunct_3,e2_56_1__rfunct_3)]]). fof(dt_c2_56_1__rfunct_3,plain,( m1_subset_1(c2_56_1__rfunct_3,c1_56__rfunct_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_finseq_1,cc1_xreal_0,cc2_xreal_0,cc3_nat_1,cc7_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_xreal_0,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,t1_subset,cc15_membered,cc1_finset_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,dt_m1_subset_1,dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,de_c2_56_1__rfunct_3,e2_56_1__rfunct_3]), [interesting(0.65),file(rfunct_3,c2_56_1__rfunct_3),[file(rfunct_3,c2_56_1__rfunct_3)]]). fof(commutativity_k2_square_1,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => k2_square_1(A,B) = k2_square_1(B,A) ) ), file(square_1,k2_square_1), [interesting(0.9),axiom,file(square_1,k2_square_1)]). fof(idempotence_k2_square_1,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => k2_square_1(A,A) = A ) ), file(square_1,k2_square_1), [interesting(0.9),axiom,file(square_1,k2_square_1)]). fof(dt_k2_square_1,axiom,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => v1_xreal_0(k2_square_1(A,B)) ) ), file(square_1,k2_square_1), [interesting(0.9),axiom,file(square_1,k2_square_1)]). fof(commutativity_k4_square_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_square_1(A,B) = k4_square_1(B,A) ) ), file(square_1,k4_square_1), [interesting(0.9),axiom,file(square_1,k4_square_1)]). fof(idempotence_k4_square_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_square_1(A,A) = A ) ), file(square_1,k4_square_1), [interesting(0.9),axiom,file(square_1,k4_square_1)]). fof(redefinition_k4_square_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => k4_square_1(A,B) = k2_square_1(A,B) ) ), file(square_1,k4_square_1), [interesting(0.9),axiom,file(square_1,k4_square_1)]). fof(dt_k4_square_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k1_numbers) & m1_subset_1(B,k1_numbers) ) => m1_subset_1(k4_square_1(A,B),k1_numbers) ) ), file(square_1,k4_square_1), [interesting(0.9),axiom,file(square_1,k4_square_1)]). fof(dt_k2_rfunct_3,axiom,( ! [A] : ( v1_xreal_0(A) => m1_subset_1(k2_rfunct_3(A),k1_numbers) ) ), file(rfunct_3,k2_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,k2_rfunct_3)]). fof(d1_rfunct_3,definition,( ! [A] : ( v1_xreal_0(A) => k2_rfunct_3(A) = k2_square_1(A,0) ) ), file(rfunct_3,d1_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,d1_rfunct_3)]). fof(d10_rfunct_3,definition,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => ! [C] : ( ( v1_funct_1(C) & m2_relset_1(C,A,k1_numbers) ) => ( C = k19_rfunct_3(A,B) <=> ( k4_relset_1(A,k1_numbers,C) = k4_relset_1(A,k1_numbers,B) & ! [D] : ( m1_subset_1(D,A) => ( r2_hidden(D,k4_relset_1(A,k1_numbers,C)) => k2_seq_1(A,k1_numbers,C,D) = k2_rfunct_3(k2_seq_1(A,k1_numbers,B,D)) ) ) ) ) ) ) ) ), file(rfunct_3,d10_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,d10_rfunct_3)]). fof(e1_56__rfunct_3,plain,( k4_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3)) = k4_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3])],[dt_k5_ordinal2,fc5_membered,rc2_finset_1,rc2_nat_1,rc3_nat_1,reflexivity_r1_tarski,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k5_numbers,dt_m2_subset_1,cc1_finseq_1,cc1_nat_1,cc1_xreal_0,cc2_nat_1,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,cc9_membered,fc17_finseq_1,fc1_seq_1,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_numerals,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_square_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc4_membered,cc6_membered,cc7_xreal_0,fc14_finset_1,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc4_finset_1,spc0_boole,t3_subset,t4_subset,t5_subset,spc0_numerals,spc0_boole,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,d1_rfunct_3,d10_rfunct_3]), [interesting(0.8),file(rfunct_3,e1_56__rfunct_3),[file(rfunct_3,e1_56__rfunct_3)]]). fof(d13_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B,C] : ( C = k10_relat_1(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,k1_relat_1(A)) & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ), file(funct_1,d13_funct_1), [interesting(0.9),axiom,file(funct_1,d13_funct_1)]). fof(e3_56_1__rfunct_3,plain, ( r2_hidden(c2_56_1__rfunct_3,k4_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3)) & r2_hidden(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3),k2_limfunc1(0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc17_finseq_1,fc18_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,rc2_partfun1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k11_relset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,dt_k10_relat_1,dt_k11_relset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_limfunc1,dt_k2_seq_1,dt_k4_relset_1,dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e1_56_1__rfunct_3,d13_funct_1]), [interesting(0.65),file(rfunct_3,e3_56_1__rfunct_3),[file(rfunct_3,e3_56_1__rfunct_3)]]). fof(e1_56_1_1__rfunct_3,plain,( k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c2_56_1__rfunct_3) = k2_rfunct_3(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[reflexivity_r1_tarski,dt_k5_ordinal2,cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc17_finseq_1,fc1_seq_1,fc5_membered,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_square_1,dt_k2_zfmisc_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_subset_1,dt_c1_56_1__rfunct_3,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc4_finset_1,t1_numerals,t3_subset,t4_subset,t5_subset,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_limfunc1,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,spc0_boole,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,d1_rfunct_3,spc0_numerals,spc0_boole,e1_56__rfunct_3,e3_56_1__rfunct_3,d10_rfunct_3]), [interesting(0.5),file(rfunct_3,e1_56_1_1__rfunct_3),[file(rfunct_3,e1_56_1_1__rfunct_3)]]). fof(e2_56_1_1__rfunct_3,plain,( k2_rfunct_3(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3)) = k4_square_1(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_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,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k2_square_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_56_1__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k4_square_1,idempotence_k4_square_1,redefinition_k2_seq_1,redefinition_k4_square_1,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_square_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,fc2_membered,d1_rfunct_3,spc0_numerals,spc0_boole]), [interesting(0.5),file(rfunct_3,e2_56_1_1__rfunct_3),[file(rfunct_3,e2_56_1_1__rfunct_3)]]). fof(e6_56_1__rfunct_3,plain,( k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c2_56_1__rfunct_3) = k4_square_1(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3),0) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[e1_56_1_1__rfunct_3,e2_56_1_1__rfunct_3]), [interesting(0.65),file(rfunct_3,e6_56_1__rfunct_3),[file(rfunct_3,e6_56_1__rfunct_3)]]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(t3_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t5_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(B) | v3_xreal_0(A) | v2_xreal_0(B) ) ) ) ) ), file(real,t5_real), [interesting(0.9),axiom,file(real,t5_real)]). fof(t6_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) => ( v1_xboole_0(A) | v2_xreal_0(B) | v3_xreal_0(A) ) ) ) ) ), file(real,t6_real), [interesting(0.9),axiom,file(real,t6_real)]). fof(t7_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v2_xreal_0(A) & ~ v3_xreal_0(B) ) ) ) ), file(real,t7_real), [interesting(0.9),axiom,file(real,t7_real)]). fof(t8_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( ~ r1_xreal_0(A,B) & ~ v3_xreal_0(B) & ~ v2_xreal_0(A) ) ) ) ), file(real,t8_real), [interesting(0.9),axiom,file(real,t8_real)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t4_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v2_xreal_0(B) & v2_xreal_0(A) ) ) ) ), file(real,t4_real), [interesting(0.9),axiom,file(real,t4_real)]). fof(reflexivity_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => r1_xreal_0(A,A) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(connectedness_r1_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( r1_xreal_0(A,B) | r1_xreal_0(B,A) ) ) ), file(xreal_0,r1_xreal_0), [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_0_0_rfunct_3,definition,( ! [A] : ( r2_hidden(A,a_0_0_rfunct_3) <=> ? [B] : ( m1_subset_1(B,k1_numbers) & A = B & r1_xreal_0(B,0) ) ) ), file(rfunct_3,a_0_0_rfunct_3), [interesting(0.9),axiom,file(rfunct_3,a_0_0_rfunct_3)]). fof(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(fraenkel_a_1_0_limfunc1,definition,( ! [A,B] : ( v1_xreal_0(B) => ( r2_hidden(A,a_1_0_limfunc1(B)) <=> ? [C] : ( m1_subset_1(C,k1_numbers) & A = C & r1_xreal_0(C,B) ) ) ) ), file(limfunc1,a_1_0_limfunc1), [interesting(0.9),axiom,file(limfunc1,a_1_0_limfunc1)]). fof(d1_limfunc1,definition,( ! [A] : ( v1_xreal_0(A) => k2_limfunc1(A) = a_1_0_limfunc1(A) ) ), file(limfunc1,d1_limfunc1), [interesting(0.9),axiom,file(limfunc1,d1_limfunc1)]). fof(e2_56__rfunct_3,plain,( k2_limfunc1(0) = a_0_0_rfunct_3 ), inference(mizar_by,[status(thm),assumptions([])],[cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_membered,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc15_membered,cc1_finset_1,cc1_nat_1,cc2_nat_1,cc3_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc2_membered,rc1_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,t1_numerals,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,dt_k2_limfunc1,cc2_xreal_0,t2_tarski,fraenkel_a_0_0_rfunct_3,fraenkel_a_1_0_limfunc1,spc0_numerals,spc0_boole,d1_limfunc1]), [interesting(0.8),file(rfunct_3,e2_56__rfunct_3),[file(rfunct_3,e2_56__rfunct_3)]]). fof(e4_56_1__rfunct_3,plain,( ? [A] : ( m1_subset_1(A,k1_numbers) & A = k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3) & r1_xreal_0(A,0) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc17_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_56_1__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k2_seq_1,redefinition_k4_relset_1,dt_k1_numbers,dt_k2_limfunc1,dt_k2_seq_1,dt_k4_relset_1,dt_m1_subset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,fc2_membered,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_rfunct_3,spc0_numerals,spc0_boole,e3_56_1__rfunct_3,e2_56__rfunct_3,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.65),file(rfunct_3,e4_56_1__rfunct_3),[file(rfunct_3,e4_56_1__rfunct_3)]]). fof(d2_square_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(B,A) => k2_square_1(A,B) = A ) & ( ~ r1_xreal_0(B,A) => k2_square_1(A,B) = B ) ) ) ) ), file(square_1,d2_square_1), [interesting(0.9),axiom,file(square_1,d2_square_1)]). fof(e5_56_1__rfunct_3,plain,( k4_square_1(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c2_56_1__rfunct_3),0) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_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,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_56_1__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,rc1_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k2_square_1,idempotence_k2_square_1,commutativity_k4_square_1,idempotence_k4_square_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_subset_1,redefinition_k2_seq_1,redefinition_k4_square_1,dt_k1_numbers,dt_k2_seq_1,dt_k2_square_1,dt_k4_square_1,dt_m1_subset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,cc2_xreal_0,fc2_membered,spc0_numerals,spc0_boole,e4_56_1__rfunct_3,d2_square_1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.65),file(rfunct_3,e5_56_1__rfunct_3),[file(rfunct_3,e5_56_1__rfunct_3)]]). fof(d1_tarski,definition,( ! [A,B] : ( B = k1_tarski(A) <=> ! [C] : ( r2_hidden(C,B) <=> C = A ) ) ), file(tarski,d1_tarski), [interesting(0.9),axiom,file(tarski,d1_tarski)]). fof(e7_56_1__rfunct_3,plain,( r2_hidden(k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c2_56_1__rfunct_3),k2_setwiseo(k1_numbers,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,fc10_membered,fc9_membered,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,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,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc11_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t3_subset,t4_subset,t5_subset,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k2_square_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c1_56_1__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc4_membered,rc1_finset_1,t1_numerals,t2_subset,t6_boole,t8_boole,commutativity_k4_square_1,idempotence_k4_square_1,antisymmetry_r2_hidden,redefinition_k2_seq_1,redefinition_k2_setwiseo,redefinition_k4_square_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k1_tarski,dt_k2_seq_1,dt_k2_setwiseo,dt_k4_square_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,fc1_finset_1,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e6_56_1__rfunct_3,e5_56_1__rfunct_3,d1_tarski]), [interesting(0.65),file(rfunct_3,e7_56_1__rfunct_3),[file(rfunct_3,e7_56_1__rfunct_3)]]). fof(e8_56_1__rfunct_3,plain,( r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[cc1_finseq_1,fc10_membered,fc9_membered,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc17_finseq_1,fc18_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finset_1,fc8_membered,rc2_partfun1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k11_relset_1,redefinition_k2_seq_1,redefinition_k2_setwiseo,redefinition_k4_relset_1,dt_k10_relat_1,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_limfunc1,dt_k2_seq_1,dt_k2_setwiseo,dt_k4_relset_1,dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,dt_c2_56_1__rfunct_3,de_c2_56_1__rfunct_3,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e7_56_1__rfunct_3,e1_56__rfunct_3,e3_56_1__rfunct_3,d13_funct_1]), [interesting(0.65),file(rfunct_3,e8_56_1__rfunct_3),[file(rfunct_3,e8_56_1__rfunct_3)]]). fof(i3_56_1__rfunct_3,theorem,( $true ), introduced(tautology,[file(rfunct_3,i3_56_1__rfunct_3)]), [interesting(0.65),trivial,file(rfunct_3,i3_56_1__rfunct_3)]). fof(i2_56_1__rfunct_3,plain,( r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ), inference(conclusion,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3,e1_56_1__rfunct_3])],[e8_56_1__rfunct_3,i3_56_1__rfunct_3]), [interesting(0.65),file(rfunct_3,i2_56_1__rfunct_3),[file(rfunct_3,i2_56_1__rfunct_3)]]). fof(i1_56_1__rfunct_3,plain,( ~ ( r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) & ~ r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c1_56_1__rfunct_3,dt_c2_56__rfunct_3]),discharge_asm(discharge,[e1_56_1__rfunct_3])],[e1_56_1__rfunct_3,i2_56_1__rfunct_3]), [interesting(0.65),file(rfunct_3,i1_56_1__rfunct_3),[file(rfunct_3,i1_56_1__rfunct_3)]]). fof(i1_56_1_tmp__rfunct_3,plain,( ~ ( r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) & ~ r2_hidden(c1_56_1__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3]),discharge_asm(discharge,[dt_c1_56_1__rfunct_3])],[dt_c1_56_1__rfunct_3,i1_56_1__rfunct_3]), [interesting(0.8),e3_56__rfunct_3]). fof(e3_56__rfunct_3,plain,( r1_tarski(k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0)),k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ), inference(let,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3])],[i1_56_1_tmp__rfunct_3,fc10_membered,fc9_membered,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc18_finseq_1,fc5_membered,fc7_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_relat_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finset_1,fc8_membered,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k11_relset_1,redefinition_k2_setwiseo,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_limfunc1,dt_k2_setwiseo,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,fc2_membered,spc0_numerals,spc0_boole,d3_tarski,dh_c1_56_1__rfunct_3]), [interesting(0.8),file(rfunct_3,e3_56__rfunct_3),[file(rfunct_3,e3_56__rfunct_3)]]). fof(dt_c4_56__rfunct_3,assumption,( $true ), introduced(assumption,[file(rfunct_3,c4_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c4_56__rfunct_3)]). fof(dh_c4_56__rfunct_3,definition, ( ~ ( r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) & ~ r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ) => ! [A] : ~ ( r2_hidden(A,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) & ~ r2_hidden(A,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ) ), introduced(definition,[new_symbol(c4_56__rfunct_3),file(rfunct_3,c4_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c4_56__rfunct_3)]). fof(e4_56__rfunct_3,assumption,( r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) ), introduced(assumption,[file(rfunct_3,e4_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,e4_56__rfunct_3)]). fof(de_c5_56__rfunct_3,definition,( c5_56__rfunct_3 = c4_56__rfunct_3 ), introduced(definition,[new_symbol(c5_56__rfunct_3),file(rfunct_3,c5_56__rfunct_3)]), [interesting(0.8),axiom,file(rfunct_3,c5_56__rfunct_3)]). fof(e5_56__rfunct_3,plain,( m1_subset_1(c4_56__rfunct_3,c1_56__rfunct_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,e4_56__rfunct_3])],[cc1_finseq_1,fc10_membered,fc9_membered,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_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,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc18_finseq_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_relat_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_finset_1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k11_relset_1,redefinition_k2_setwiseo,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_setwiseo,dt_m1_subset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e4_56__rfunct_3]), [interesting(0.8),file(rfunct_3,e5_56__rfunct_3),[file(rfunct_3,e5_56__rfunct_3)]]). fof(dt_c5_56__rfunct_3,plain,( m1_subset_1(c5_56__rfunct_3,c1_56__rfunct_3) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,e4_56__rfunct_3])],[cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc8_xreal_0,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,cc1_finseq_1,cc1_xreal_0,cc2_xreal_0,cc3_nat_1,cc7_xreal_0,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_xreal_0,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc2_finseq_1,fc6_membered,rc1_finset_1,rc1_membered,t1_subset,cc15_membered,cc1_finset_1,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_subset_1,dt_m1_subset_1,dt_c1_56__rfunct_3,dt_c4_56__rfunct_3,de_c5_56__rfunct_3,e5_56__rfunct_3]), [interesting(0.8),file(rfunct_3,c5_56__rfunct_3),[file(rfunct_3,c5_56__rfunct_3)]]). fof(e6_56__rfunct_3,plain, ( r2_hidden(c5_56__rfunct_3,k4_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3)) & r2_hidden(k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c5_56__rfunct_3),k2_setwiseo(k1_numbers,0)) ), inference(mizar_by,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[cc1_finseq_1,fc10_membered,fc9_membered,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_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,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc17_finseq_1,fc18_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,fc1_finset_1,rc2_partfun1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k11_relset_1,redefinition_k2_seq_1,redefinition_k2_setwiseo,redefinition_k4_relset_1,dt_k10_relat_1,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_seq_1,dt_k2_setwiseo,dt_k4_relset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e1_56__rfunct_3,e4_56__rfunct_3,d13_funct_1]), [interesting(0.8),file(rfunct_3,e6_56__rfunct_3),[file(rfunct_3,e6_56__rfunct_3)]]). fof(e1_56_2__rfunct_3,plain,( k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c5_56__rfunct_3) = k2_rfunct_3(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c5_56__rfunct_3)) ), inference(mizar_by,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[reflexivity_r1_tarski,dt_k5_ordinal2,cc1_finseq_1,cc1_xreal_0,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_membered,fc11_membered,fc17_finseq_1,fc1_seq_1,fc5_membered,fc7_membered,fc9_membered,rc1_finseq_1,rc1_nat_1,rc1_partfun1,rc1_seq_1,rc1_xreal_0,rc2_finseq_1,rc2_finset_1,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finseq_1,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_square_1,dt_k2_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m2_subset_1,dt_c4_56__rfunct_3,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_membered,cc1_nat_1,cc1_relset_1,cc1_seq_1,cc20_membered,cc2_finset_1,cc2_membered,cc2_nat_1,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc14_finset_1,fc1_finset_1,fc2_finseq_1,fc6_membered,fc8_membered,rc1_finset_1,rc1_membered,rc3_finset_1,rc4_finset_1,t1_numerals,t3_subset,t4_subset,t5_subset,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_relset_1,redefinition_k2_seq_1,redefinition_k2_setwiseo,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k2_setwiseo,dt_k4_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,cc15_membered,cc1_finset_1,fc2_membered,spc0_boole,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole,d1_rfunct_3,spc0_numerals,spc0_boole,e1_56__rfunct_3,e6_56__rfunct_3,d10_rfunct_3]), [interesting(0.65),file(rfunct_3,e1_56_2__rfunct_3),[file(rfunct_3,e1_56_2__rfunct_3)]]). fof(e2_56_2__rfunct_3,plain,( k2_rfunct_3(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c5_56__rfunct_3)) = k4_square_1(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c5_56__rfunct_3),0) ), inference(mizar_by,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,e4_56__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_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,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_square_1,idempotence_k2_square_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k2_square_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c4_56__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,t1_numerals,t2_subset,t6_boole,t7_boole,t8_boole,commutativity_k4_square_1,idempotence_k4_square_1,redefinition_k2_seq_1,redefinition_k4_square_1,dt_k1_numbers,dt_k2_rfunct_3,dt_k2_seq_1,dt_k4_square_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,fc2_membered,d1_rfunct_3,spc0_numerals,spc0_boole]), [interesting(0.65),file(rfunct_3,e2_56_2__rfunct_3),[file(rfunct_3,e2_56_2__rfunct_3)]]). fof(e8_56__rfunct_3,plain,( k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c5_56__rfunct_3) = k4_square_1(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c5_56__rfunct_3),0) ), inference(iterative_eq,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,e4_56__rfunct_3])],[e1_56_2__rfunct_3,e2_56_2__rfunct_3]), [interesting(0.8),file(rfunct_3,e8_56__rfunct_3),[file(rfunct_3,e8_56__rfunct_3)]]). fof(e7_56__rfunct_3,plain,( k2_seq_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),c5_56__rfunct_3) = 0 ), inference(mizar_by,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[cc1_finseq_1,fc10_membered,fc9_membered,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_membered,cc2_xreal_0,cc3_membered,cc3_nat_1,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc17_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,fc8_membered,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c4_56__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_finset_1,cc2_nat_1,cc4_membered,cc6_membered,cc9_membered,rc1_finset_1,rc3_finset_1,rc4_finset_1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k2_seq_1,redefinition_k2_setwiseo,redefinition_k4_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k1_tarski,dt_k2_seq_1,dt_k2_setwiseo,dt_k4_relset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,fc1_finset_1,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e6_56__rfunct_3,d1_tarski]), [interesting(0.8),file(rfunct_3,e7_56__rfunct_3),[file(rfunct_3,e7_56__rfunct_3)]]). fof(e9_56__rfunct_3,plain,( r1_xreal_0(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c5_56__rfunct_3),0) ), inference(mizar_by,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[reflexivity_r1_tarski,cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_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,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc8_xreal_0,cc9_membered,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c4_56__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc3_xreal_0,cc4_membered,cc6_xreal_0,cc7_xreal_0,rc1_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k2_square_1,idempotence_k2_square_1,commutativity_k4_square_1,idempotence_k4_square_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k2_seq_1,redefinition_k4_square_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_seq_1,dt_k2_square_1,dt_k4_square_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,cc2_xreal_0,fc2_membered,spc0_numerals,spc0_boole,e8_56__rfunct_3,e7_56__rfunct_3,d2_square_1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.8),file(rfunct_3,e9_56__rfunct_3),[file(rfunct_3,e9_56__rfunct_3)]]). fof(e10_56__rfunct_3,plain,( r2_hidden(k2_seq_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,c5_56__rfunct_3),k2_limfunc1(0)) ), inference(mizar_by,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[cc1_finseq_1,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc14_finset_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,dt_c4_56__rfunct_3,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k2_seq_1,dt_k1_numbers,dt_k2_limfunc1,dt_k2_seq_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,fc2_membered,t1_subset,t7_boole,t2_tarski,fraenkel_a_0_0_rfunct_3,spc0_numerals,spc0_boole,e9_56__rfunct_3,e2_56__rfunct_3,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.8),file(rfunct_3,e10_56__rfunct_3),[file(rfunct_3,e10_56__rfunct_3)]]). fof(e11_56__rfunct_3,plain,( r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ), inference(mizar_by,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[cc1_finseq_1,fc10_membered,fc9_membered,rc1_finseq_1,rc1_partfun1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc6_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc17_finseq_1,fc18_finseq_1,fc1_seq_1,fc2_finseq_1,fc5_membered,fc6_membered,fc7_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finset_1,fc8_membered,rc2_partfun1,t1_numerals,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k11_relset_1,redefinition_k2_seq_1,redefinition_k2_setwiseo,redefinition_k4_relset_1,dt_k10_relat_1,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_funct_1,dt_k1_numbers,dt_k1_relat_1,dt_k2_limfunc1,dt_k2_seq_1,dt_k2_setwiseo,dt_k4_relset_1,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,dt_c4_56__rfunct_3,dt_c5_56__rfunct_3,de_c5_56__rfunct_3,fc2_membered,t1_subset,t7_boole,spc0_numerals,spc0_boole,e10_56__rfunct_3,e6_56__rfunct_3,d13_funct_1]), [interesting(0.8),file(rfunct_3,e11_56__rfunct_3),[file(rfunct_3,e11_56__rfunct_3)]]). fof(i6_56__rfunct_3,theorem,( $true ), introduced(tautology,[file(rfunct_3,i6_56__rfunct_3)]), [interesting(0.8),trivial,file(rfunct_3,i6_56__rfunct_3)]). fof(i5_56__rfunct_3,plain,( r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ), inference(conclusion,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,e4_56__rfunct_3])],[e11_56__rfunct_3,i6_56__rfunct_3]), [interesting(0.8),file(rfunct_3,i5_56__rfunct_3),[file(rfunct_3,i5_56__rfunct_3)]]). fof(i4_56__rfunct_3,plain,( ~ ( r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) & ~ r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c4_56__rfunct_3,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3]),discharge_asm(discharge,[e4_56__rfunct_3])],[e4_56__rfunct_3,i5_56__rfunct_3]), [interesting(0.8),file(rfunct_3,i4_56__rfunct_3),[file(rfunct_3,i4_56__rfunct_3)]]). fof(i4_56_tmp__rfunct_3,plain,( ~ ( r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0))) & ~ r2_hidden(c4_56__rfunct_3,k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3]),discharge_asm(discharge,[dt_c4_56__rfunct_3])],[dt_c4_56__rfunct_3,i4_56__rfunct_3]), [interesting(0.8),i3_56__rfunct_3]). fof(i3_56__rfunct_3,plain,( r1_tarski(k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0)),k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0))) ), inference(let,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3])],[i4_56_tmp__rfunct_3,fc10_membered,fc9_membered,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc18_finseq_1,fc5_membered,fc7_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_relat_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finset_1,fc8_membered,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k11_relset_1,redefinition_k2_setwiseo,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_limfunc1,dt_k2_setwiseo,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,fc2_membered,spc0_numerals,spc0_boole,d3_tarski,dh_c4_56__rfunct_3]), [interesting(0.8),file(rfunct_3,i3_56__rfunct_3),[file(rfunct_3,i3_56__rfunct_3)]]). fof(i2_56__rfunct_3,plain,( k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0)) ), inference(conclusion,[status(thm),assumptions([dt_c1_56__rfunct_3,dt_c2_56__rfunct_3])],[fc10_membered,fc9_membered,dt_k2_zfmisc_1,dt_k5_ordinal2,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_membered,cc1_relset_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc3_membered,cc3_nat_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc11_membered,fc14_finset_1,fc18_finseq_1,fc5_membered,fc7_membered,rc1_finset_1,rc1_membered,rc1_nat_1,rc1_seq_1,rc1_xreal_0,rc2_nat_1,rc2_partfun1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,redefinition_k5_numbers,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k10_relat_1,dt_k1_tarski,dt_k1_zfmisc_1,dt_k5_finsub_1,dt_k5_numbers,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_nat_1,cc1_seq_1,cc2_nat_1,cc2_xreal_0,cc3_xreal_0,cc4_membered,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc9_membered,fc1_finset_1,fc8_membered,reflexivity_r1_tarski,redefinition_k11_relset_1,redefinition_k2_setwiseo,dt_k11_relset_1,dt_k19_rfunct_3,dt_k1_numbers,dt_k2_limfunc1,dt_k2_setwiseo,dt_c1_56__rfunct_3,dt_c2_56__rfunct_3,fc2_membered,spc0_numerals,spc0_boole,d10_xboole_0,e3_56__rfunct_3,i3_56__rfunct_3]), [interesting(0.8),file(rfunct_3,i2_56__rfunct_3),[file(rfunct_3,i2_56__rfunct_3)]]). fof(i2_56_tmp__rfunct_3,plain, ( ( v1_funct_1(c2_56__rfunct_3) & m2_relset_1(c2_56__rfunct_3,c1_56__rfunct_3,k1_numbers) ) => k11_relset_1(c1_56__rfunct_3,k1_numbers,c2_56__rfunct_3,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,c2_56__rfunct_3),k2_setwiseo(k1_numbers,0)) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_56__rfunct_3]),discharge_asm(discharge,[dt_c2_56__rfunct_3])],[dt_c2_56__rfunct_3,i2_56__rfunct_3]), [interesting(0.8),i1_56__rfunct_3]). fof(i1_56__rfunct_3,plain,( ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_56__rfunct_3,k1_numbers) ) => k11_relset_1(c1_56__rfunct_3,k1_numbers,A,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,A),k2_setwiseo(k1_numbers,0)) ) ), inference(let,[status(thm),assumptions([dt_c1_56__rfunct_3])],[i2_56_tmp__rfunct_3,dh_c2_56__rfunct_3]), [interesting(0.8),file(rfunct_3,i1_56__rfunct_3),[file(rfunct_3,i1_56__rfunct_3)]]). fof(i1_56_tmp__rfunct_3,plain, ( ~ v1_xboole_0(c1_56__rfunct_3) => ! [A] : ( ( v1_funct_1(A) & m2_relset_1(A,c1_56__rfunct_3,k1_numbers) ) => k11_relset_1(c1_56__rfunct_3,k1_numbers,A,k2_limfunc1(0)) = k11_relset_1(c1_56__rfunct_3,k1_numbers,k19_rfunct_3(c1_56__rfunct_3,A),k2_setwiseo(k1_numbers,0)) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_56__rfunct_3])],[dt_c1_56__rfunct_3,i1_56__rfunct_3]), [interesting(1),t39_rfunct_3]). fof(t39_rfunct_3,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ( v1_funct_1(B) & m2_relset_1(B,A,k1_numbers) ) => k11_relset_1(A,k1_numbers,B,k2_limfunc1(0)) = k11_relset_1(A,k1_numbers,k19_rfunct_3(A,B),k2_setwiseo(k1_numbers,0)) ) ) ), inference(let,[status(thm),assumptions([])],[i1_56_tmp__rfunct_3,dh_c1_56__rfunct_3]), [interesting(1),file(rfunct_3,t39_rfunct_3),[file(rfunct_3,t39_rfunct_3)]]).