% Mizar ND problem: t6_card_4,card_4,88,36 fof(dh_c1_5__card_4,definition, ( ( m2_subset_1(c1_5__card_4,k1_numbers,k5_numbers) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5__card_4) => A = c1_5__card_4 ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( v3_ordinal1(C) => ( r2_wellord2(C,B) => C = B ) ) ) ), introduced(definition,[new_symbol(c1_5__card_4),file(card_4,c1_5__card_4)]), [interesting(0.8),axiom,file(card_4,c1_5__card_4)]). fof(dh_c2_5__card_4,definition, ( ( v3_ordinal1(c2_5__card_4) => ( r2_wellord2(c2_5__card_4,c1_5__card_4) => c2_5__card_4 = c1_5__card_4 ) ) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5__card_4) => A = c1_5__card_4 ) ) ), introduced(definition,[new_symbol(c2_5__card_4),file(card_4,c2_5__card_4)]), [interesting(0.8),axiom,file(card_4,c2_5__card_4)]). fof(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(cc2_funct_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ) ), file(funct_1,cc2_funct_1), [interesting(0.9),axiom,file(funct_1,cc2_funct_1)]). fof(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(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc1_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc1_funct_1), [interesting(0.9),axiom,file(funct_1,rc1_funct_1)]). fof(rc2_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_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_xboole_0(A) & v1_funct_1(A) ) ), file(funct_1,rc2_funct_1), [interesting(0.9),axiom,file(funct_1,rc2_funct_1)]). fof(rc2_ordinal1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc2_ordinal1), [interesting(0.9),axiom,file(ordinal1,rc2_ordinal1)]). 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_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) ) ), file(funct_1,rc3_funct_1), [interesting(0.9),axiom,file(funct_1,rc3_funct_1)]). fof(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_funct_1,theorem,( ? [A] : ( v1_relat_1(A) & v3_relat_1(A) & v1_funct_1(A) ) ), file(funct_1,rc4_funct_1), [interesting(0.9),axiom,file(funct_1,rc4_funct_1)]). fof(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(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(rc7_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc7_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc7_finseq_1)]). fof(rc8_finseq_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc8_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc8_finseq_1)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). fof(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(cc12_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_rat_1(B) ) ) ) ), file(membered,cc12_membered), [interesting(0.9),axiom,file(membered,cc12_membered)]). fof(cc13_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc13_membered), [interesting(0.9),axiom,file(membered,cc13_membered)]). fof(cc14_membered,theorem,( ! [A] : ( v5_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v4_ordinal2(B) & v1_xreal_0(B) & v1_int_1(B) & v1_rat_1(B) ) ) ) ), file(membered,cc14_membered), [interesting(0.9),axiom,file(membered,cc14_membered)]). fof(cc18_membered,theorem,( ! [A] : ( v3_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) ) ) ) ), file(membered,cc18_membered), [interesting(0.9),axiom,file(membered,cc18_membered)]). fof(cc19_membered,theorem,( ! [A] : ( v4_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) & v3_membered(B) & v4_membered(B) ) ) ) ), file(membered,cc19_membered), [interesting(0.9),axiom,file(membered,cc19_membered)]). fof(cc1_card_1,theorem,( ! [A] : ( v1_card_1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(card_1,cc1_card_1), [interesting(0.9),axiom,file(card_1,cc1_card_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_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(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_membered,theorem,( ! [A] : ( v3_membered(A) => v2_membered(A) ) ), file(membered,cc3_membered), [interesting(0.9),axiom,file(membered,cc3_membered)]). 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(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(fc2_ordinal1,theorem, ( 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_xboole_0(k1_xboole_0) & v1_ordinal1(k1_xboole_0) & v2_ordinal1(k1_xboole_0) & v3_ordinal1(k1_xboole_0) ), file(ordinal1,fc2_ordinal1), [interesting(0.9),axiom,file(ordinal1,fc2_ordinal1)]). fof(fc6_membered,theorem, ( v1_xboole_0(k1_xboole_0) & v1_membered(k1_xboole_0) & v2_membered(k1_xboole_0) & v3_membered(k1_xboole_0) & v4_membered(k1_xboole_0) & v5_membered(k1_xboole_0) ), file(membered,fc6_membered), [interesting(0.9),axiom,file(membered,fc6_membered)]). fof(rc1_card_1,theorem,( ? [A] : v1_card_1(A) ), file(card_1,rc1_card_1), [interesting(0.9),axiom,file(card_1,rc1_card_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_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_card_1,theorem,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v1_finset_1(A) & v1_card_1(A) ) ), file(card_1,rc2_card_1), [interesting(0.9),axiom,file(card_1,rc2_card_1)]). 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(t1_subset,theorem,( ! [A,B] : ( r2_hidden(A,B) => m1_subset_1(A,B) ) ), file(subset,t1_subset), [interesting(0.9),axiom,file(subset,t1_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(cc10_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,A) => v1_xcmplx_0(B) ) ) ), file(membered,cc10_membered), [interesting(0.9),axiom,file(membered,cc10_membered)]). fof(cc11_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,A) => ( v1_xcmplx_0(B) & v1_xreal_0(B) ) ) ) ), file(membered,cc11_membered), [interesting(0.9),axiom,file(membered,cc11_membered)]). fof(cc15_membered,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_membered(A) & v2_membered(A) & v3_membered(A) & v4_membered(A) & v5_membered(A) ) ) ), file(membered,cc15_membered), [interesting(0.9),axiom,file(membered,cc15_membered)]). fof(cc16_membered,theorem,( ! [A] : ( v1_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => v1_membered(B) ) ) ), file(membered,cc16_membered), [interesting(0.9),axiom,file(membered,cc16_membered)]). fof(cc17_membered,theorem,( ! [A] : ( v2_membered(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ( v1_membered(B) & v2_membered(B) ) ) ) ), file(membered,cc17_membered), [interesting(0.9),axiom,file(membered,cc17_membered)]). fof(cc1_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_funct_1,theorem,( ! [A] : ( v1_xboole_0(A) => v1_funct_1(A) ) ), file(funct_1,cc1_funct_1), [interesting(0.9),axiom,file(funct_1,cc1_funct_1)]). fof(cc2_card_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc2_card_1), [interesting(0.9),axiom,file(card_1,cc2_card_1)]). fof(cc2_ordinal1,theorem,( ! [A] : ( ( v1_ordinal1(A) & v2_ordinal1(A) ) => v3_ordinal1(A) ) ), file(ordinal1,cc2_ordinal1), [interesting(0.9),axiom,file(ordinal1,cc2_ordinal1)]). fof(cc3_card_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_finset_1(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_card_1(A) ) ) ), file(card_1,cc3_card_1), [interesting(0.9),axiom,file(card_1,cc3_card_1)]). fof(cc3_ordinal1,theorem,( ! [A] : ( v1_xboole_0(A) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ) ), file(ordinal1,cc3_ordinal1), [interesting(0.9),axiom,file(ordinal1,cc3_ordinal1)]). 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(fc1_ordinal2,theorem, ( v1_ordinal1(k5_ordinal2) & v2_ordinal1(k5_ordinal2) & v3_ordinal1(k5_ordinal2) & ~ v1_xboole_0(k5_ordinal2) ), file(ordinal2,fc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,fc1_ordinal2)]). fof(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_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(rc1_ordinal1,theorem,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc1_ordinal1), [interesting(0.9),axiom,file(ordinal1,rc1_ordinal1)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(rc3_ordinal1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) ) ), file(ordinal1,rc3_ordinal1), [interesting(0.9),axiom,file(ordinal1,rc3_ordinal1)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(t8_boole,theorem,( ! [A,B] : ~ ( v1_xboole_0(A) & A != B & v1_xboole_0(B) ) ), file(boole,t8_boole), [interesting(0.9),axiom,file(boole,t8_boole)]). fof(symmetry_r2_wellord2,theorem,( ! [A,B] : ( r2_wellord2(A,B) => r2_wellord2(B,A) ) ), file(wellord2,r2_wellord2), [interesting(0.9),axiom,file(wellord2,r2_wellord2)]). fof(reflexivity_r2_wellord2,theorem,( ! [A,B] : r2_wellord2(A,A) ), file(wellord2,r2_wellord2), [interesting(0.9),axiom,file(wellord2,r2_wellord2)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_k5_numbers,definition,( k5_numbers = k5_ordinal2 ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_r2_wellord2,definition,( ! [A,B] : ( r2_wellord2(A,B) <=> r2_tarski(A,B) ) ), file(wellord2,r2_wellord2), [interesting(0.9),axiom,file(wellord2,r2_wellord2)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). fof(dt_k5_numbers,axiom,( m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) ), file(numbers,k5_numbers), [interesting(0.9),axiom,file(numbers,k5_numbers)]). fof(dt_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) => m1_subset_1(C,A) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_c1_5__card_4,assumption,( m2_subset_1(c1_5__card_4,k1_numbers,k5_numbers) ), introduced(assumption,[file(card_4,c1_5__card_4)]), [interesting(0.8),axiom,file(card_4,c1_5__card_4)]). fof(dt_c2_5__card_4,assumption,( v3_ordinal1(c2_5__card_4) ), introduced(assumption,[file(card_4,c2_5__card_4)]), [interesting(0.8),axiom,file(card_4,c2_5__card_4)]). fof(cc1_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ( v1_ordinal1(A) & v2_ordinal1(A) ) ) ), file(ordinal1,cc1_ordinal1), [interesting(0.9),axiom,file(ordinal1,cc1_ordinal1)]). 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(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(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(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc8_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc8_xreal_0)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(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_k1_nat_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k1_nat_1(B,A) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(redefinition_k1_nat_1,definition,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => k1_nat_1(A,B) = k2_xcmplx_0(A,B) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_k1_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(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(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc0_boole,theorem,( v1_xboole_0(0) ), file(boole,spc0_boole), [interesting(0.9),axiom,file(boole,spc0_boole)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(s1_nat_1__e3_5__card_4,theorem, ( ( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,0) => A = 0 ) ) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ! [C] : ( v3_ordinal1(C) => ( r2_wellord2(C,B) => C = B ) ) => ! [D] : ( v3_ordinal1(D) => ( r2_wellord2(D,k1_nat_1(B,1)) => D = k1_nat_1(B,1) ) ) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [E] : ( v3_ordinal1(E) => ( r2_wellord2(E,B) => E = B ) ) ) ), file(card_4,s1_nat_1__e3_5__card_4), [interesting(0.9),axiom,file(card_4,s1_nat_1__e3_5__card_4)]). fof(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t46_card_1,theorem,( ! [A] : ( r2_wellord2(A,k1_xboole_0) <=> A = k1_xboole_0 ) ), file(card_1,t46_card_1), [interesting(0.9),axiom,file(card_1,t46_card_1)]). fof(e1_5__card_4,plain,( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,0) => A = 0 ) ) ), inference(mizar_by,[status(thm),assumptions([])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_zfmisc_1,dt_k5_ordinal2,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_card_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc1_ordinal2,fc1_subset_1,fc5_membered,rc1_card_1,rc1_subset_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc2_card_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,fc2_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc3_ordinal1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_numerals,t2_subset,t7_boole,t8_boole,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_r2_wellord2,dt_k1_xboole_0,cc1_ordinal1,fc2_finseq_1,fc2_ordinal1,fc6_membered,t6_boole,spc0_numerals,spc0_boole,t46_card_1]), [interesting(0.8),file(card_4,e1_5__card_4),[file(card_4,e1_5__card_4)]]). fof(dh_c1_5_1__card_4,definition, ( ( m2_subset_1(c1_5_1__card_4,k1_numbers,k5_numbers) => ( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5_1__card_4) => A = c1_5_1__card_4 ) ) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,k1_nat_1(c1_5_1__card_4,1)) => A = k1_nat_1(c1_5_1__card_4,1) ) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ! [C] : ( v3_ordinal1(C) => ( r2_wellord2(C,B) => C = B ) ) => ! [C] : ( v3_ordinal1(C) => ( r2_wellord2(C,k1_nat_1(B,1)) => C = k1_nat_1(B,1) ) ) ) ) ), introduced(definition,[new_symbol(c1_5_1__card_4),file(card_4,c1_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,c1_5_1__card_4)]). fof(e1_5_1__card_4,assumption,( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5_1__card_4) => A = c1_5_1__card_4 ) ) ), introduced(assumption,[file(card_4,e1_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,e1_5_1__card_4)]). fof(dh_c2_5_1__card_4,definition, ( ( v3_ordinal1(c2_5_1__card_4) => ( r2_wellord2(c2_5_1__card_4,k1_nat_1(c1_5_1__card_4,1)) => c2_5_1__card_4 = k1_nat_1(c1_5_1__card_4,1) ) ) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,k1_nat_1(c1_5_1__card_4,1)) => A = k1_nat_1(c1_5_1__card_4,1) ) ) ), introduced(definition,[new_symbol(c2_5_1__card_4),file(card_4,c2_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,c2_5_1__card_4)]). fof(e2_5_1__card_4,assumption,( r2_wellord2(c2_5_1__card_4,k1_nat_1(c1_5_1__card_4,1)) ), introduced(assumption,[file(card_4,e2_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,e2_5_1__card_4)]). 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(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(fc13_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc13_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc13_xreal_0)]). fof(fc14_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc14_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc14_xreal_0)]). fof(fc15_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc15_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc15_xreal_0)]). fof(fc16_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc16_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc16_xreal_0)]). fof(fc17_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v2_xreal_0(k6_xcmplx_0(A,B)) & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc17_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc17_xreal_0)]). fof(fc18_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & ~ v2_xreal_0(k6_xcmplx_0(B,A)) & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc18_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc18_xreal_0)]). fof(fc19_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(A,B)) & v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & ~ v2_xreal_0(k6_xcmplx_0(A,B)) & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc19_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc19_xreal_0)]). fof(fc1_xreal_0,theorem,( ! [A] : ( v1_xreal_0(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) ) ) ), file(xreal_0,fc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc1_xreal_0)]). fof(fc20_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k6_xcmplx_0(B,A)) & v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v2_xreal_0(k6_xcmplx_0(B,A)) & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc20_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc20_xreal_0)]). fof(fc5_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc5_xreal_0)]). fof(fc8_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(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(fc12_finset_1,theorem,( ! [A,B] : ( v1_finset_1(A) => v1_finset_1(k4_xboole_0(A,B)) ) ), file(finset_1,fc12_finset_1), [interesting(0.9),axiom,file(finset_1,fc12_finset_1)]). fof(fc37_membered,theorem,( ! [A,B] : ( v1_membered(A) => v1_membered(k4_xboole_0(A,B)) ) ), file(membered,fc37_membered), [interesting(0.9),axiom,file(membered,fc37_membered)]). fof(fc38_membered,theorem,( ! [A,B] : ( v2_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc38_membered), [interesting(0.9),axiom,file(membered,fc38_membered)]). fof(fc39_membered,theorem,( ! [A,B] : ( v3_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc39_membered), [interesting(0.9),axiom,file(membered,fc39_membered)]). fof(fc40_membered,theorem,( ! [A,B] : ( v4_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc40_membered), [interesting(0.9),axiom,file(membered,fc40_membered)]). fof(fc41_membered,theorem,( ! [A,B] : ( v5_membered(A) => ( v1_membered(k4_xboole_0(A,B)) & v2_membered(k4_xboole_0(A,B)) & v3_membered(k4_xboole_0(A,B)) & v4_membered(k4_xboole_0(A,B)) & v5_membered(k4_xboole_0(A,B)) ) ) ), file(membered,fc41_membered), [interesting(0.9),axiom,file(membered,fc41_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(rqRealAdd__k2_xcmplx_0__r0_r0_r0,theorem,( k2_xcmplx_0(0,0) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r0_r0)]). fof(rqRealAdd__k2_xcmplx_0__r0_r1_r1,theorem,( k2_xcmplx_0(0,1) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r0_r2_r2,theorem,( k2_xcmplx_0(0,2) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_r2_r2)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,theorem,( k2_xcmplx_0(0,k4_xcmplx_0(2)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2)]). fof(rqRealAdd__k2_xcmplx_0__r1_r0_r1,theorem,( k2_xcmplx_0(1,0) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r0_r1)]). fof(rqRealAdd__k2_xcmplx_0__r1_r1_r2,theorem,( k2_xcmplx_0(1,1) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_r1_r2)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm1_r0,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm1_r0)]). fof(rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,theorem,( k2_xcmplx_0(1,k4_xcmplx_0(2)) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1)]). fof(rqRealAdd__k2_xcmplx_0__r2_r0_r2,theorem,( k2_xcmplx_0(2,0) = 2 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_r0_r2)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm1_r1,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm1_r1)]). fof(rqRealAdd__k2_xcmplx_0__r2_rm2_r0,theorem,( k2_xcmplx_0(2,k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__r2_rm2_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r1_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),1) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r1_r0)]). fof(rqRealAdd__k2_xcmplx_0__rm1_r2_r1,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),2) = 1 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_r2_r1)]). fof(rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),1) = k4_xcmplx_0(1) ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1)]). fof(rqRealAdd__k2_xcmplx_0__rm2_r2_r0,theorem,( k2_xcmplx_0(k4_xcmplx_0(2),2) = 0 ), file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealAdd__k2_xcmplx_0__rm2_r2_r0)]). fof(spc1_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(A,k4_xcmplx_0(B)) = k6_xcmplx_0(A,B) ) ), file(arithm,spc1_arithm), [interesting(0.9),axiom,file(arithm,spc1_arithm)]). fof(spc6_arithm,theorem,( ! [A,B,C] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) & v1_xcmplx_0(C) ) => k2_xcmplx_0(k2_xcmplx_0(A,B),C) = k2_xcmplx_0(A,k2_xcmplx_0(B,C)) ) ), file(arithm,spc6_arithm), [interesting(0.9),axiom,file(arithm,spc6_arithm)]). fof(spc8_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k2_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k4_xcmplx_0(k2_xcmplx_0(A,B)) ) ), file(arithm,spc8_arithm), [interesting(0.9),axiom,file(arithm,spc8_arithm)]). fof(spc9_arithm,theorem,( ! [A,B] : ( ( v1_xcmplx_0(A) & v1_xcmplx_0(B) ) => k6_xcmplx_0(k4_xcmplx_0(A),k4_xcmplx_0(B)) = k6_xcmplx_0(B,A) ) ), file(arithm,spc9_arithm), [interesting(0.9),axiom,file(arithm,spc9_arithm)]). fof(t1_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k2_xcmplx_0(A,0) = A ) ), file(arithm,t1_arithm), [interesting(0.9),axiom,file(arithm,t1_arithm)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(involutiveness_k4_xcmplx_0,theorem,( ! [A] : ( v1_xcmplx_0(A) => k4_xcmplx_0(k4_xcmplx_0(A)) = A ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(redefinition_k6_domain_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => k6_domain_1(A,B) = k1_tarski(B) ) ), file(domain_1,k6_domain_1), [interesting(0.9),axiom,file(domain_1,k6_domain_1)]). fof(dt_k1_ordinal1,axiom,( $true ), file(ordinal1,k1_ordinal1), [interesting(0.9),axiom,file(ordinal1,k1_ordinal1)]). fof(dt_k1_tarski,axiom,( $true ), file(tarski,k1_tarski), [interesting(0.9),axiom,file(tarski,k1_tarski)]). fof(dt_k4_xboole_0,axiom,( $true ), file(xboole_0,k4_xboole_0), [interesting(0.9),axiom,file(xboole_0,k4_xboole_0)]). fof(dt_k4_xcmplx_0,axiom,( ! [A] : ( v1_xcmplx_0(A) => v1_xcmplx_0(k4_xcmplx_0(A)) ) ), file(xcmplx_0,k4_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k4_xcmplx_0)]). fof(dt_k6_domain_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & m1_subset_1(B,A) ) => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) ), file(domain_1,k6_domain_1), [interesting(0.9),axiom,file(domain_1,k6_domain_1)]). fof(dt_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(dt_c1_5_1__card_4,assumption,( m2_subset_1(c1_5_1__card_4,k1_numbers,k5_numbers) ), introduced(assumption,[file(card_4,c1_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,c1_5_1__card_4)]). fof(dt_c2_5_1__card_4,assumption,( v3_ordinal1(c2_5_1__card_4) ), introduced(assumption,[file(card_4,c2_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,c2_5_1__card_4)]). fof(dh_c3_5_1__card_4,definition, ( ? [A] : ( v3_ordinal1(A) & c2_5_1__card_4 = k1_ordinal1(A) ) => ( v3_ordinal1(c3_5_1__card_4) & c2_5_1__card_4 = k1_ordinal1(c3_5_1__card_4) ) ), introduced(definition,[new_symbol(c3_5_1__card_4),file(card_4,c3_5_1__card_4)]), [interesting(0.65),axiom,file(card_4,c3_5_1__card_4)]). fof(rc1_ordinal2,theorem,( ? [A] : ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal1(A) ) ), file(ordinal2,rc1_ordinal2), [interesting(0.9),axiom,file(ordinal2,rc1_ordinal2)]). fof(fc1_ordinal1,theorem,( ! [A] : ~ v1_xboole_0(k1_ordinal1(A)) ), file(ordinal1,fc1_ordinal1), [interesting(0.9),axiom,file(ordinal1,fc1_ordinal1)]). fof(fc3_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ( ~ v1_xboole_0(k1_ordinal1(A)) & v1_ordinal1(k1_ordinal1(A)) & v2_ordinal1(k1_ordinal1(A)) & v3_ordinal1(k1_ordinal1(A)) ) ) ), file(ordinal1,fc3_ordinal1), [interesting(0.9),axiom,file(ordinal1,fc3_ordinal1)]). fof(e1_5_1_1__card_4,assumption,( v4_ordinal1(c2_5_1__card_4) ), introduced(assumption,[file(card_4,e1_5_1_1__card_4)]), [interesting(0.5),axiom,file(card_4,e1_5_1_1__card_4)]). fof(fc2_card_1,theorem,( ! [A] : ( v1_finset_1(A) => ( v1_ordinal1(k1_card_1(A)) & v2_ordinal1(k1_card_1(A)) & v3_ordinal1(k1_card_1(A)) & v1_finset_1(k1_card_1(A)) & v1_card_1(k1_card_1(A)) ) ) ), file(card_1,fc2_card_1), [interesting(0.9),axiom,file(card_1,fc2_card_1)]). fof(dt_k1_card_1,axiom,( ! [A] : v1_card_1(k1_card_1(A)) ), file(card_1,k1_card_1), [interesting(0.9),axiom,file(card_1,k1_card_1)]). fof(fc27_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc27_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc27_xreal_0)]). fof(fc28_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(B,A)) & v1_xreal_0(k7_xcmplx_0(B,A)) & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc28_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc28_xreal_0)]). fof(fc29_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc29_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc29_xreal_0)]). fof(fc30_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & v1_xreal_0(B) & ~ v2_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc30_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc30_xreal_0)]). fof(fc6_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k7_xcmplx_0(A,B)) & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc6_xreal_0)]). fof(t5_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(0,A) = 0 ) ), file(arithm,t5_arithm), [interesting(0.9),axiom,file(arithm,t5_arithm)]). fof(t6_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k7_xcmplx_0(A,1) = A ) ), file(arithm,t6_arithm), [interesting(0.9),axiom,file(arithm,t6_arithm)]). fof(reflexivity_r1_ordinal1,theorem,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => r1_ordinal1(A,A) ) ), file(ordinal1,r1_ordinal1), [interesting(0.9),axiom,file(ordinal1,r1_ordinal1)]). fof(connectedness_r1_ordinal1,theorem,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => ( r1_ordinal1(A,B) | r1_ordinal1(B,A) ) ) ), file(ordinal1,r1_ordinal1), [interesting(0.9),axiom,file(ordinal1,r1_ordinal1)]). fof(redefinition_r1_ordinal1,definition,( ! [A,B] : ( ( v3_ordinal1(A) & v3_ordinal1(B) ) => ( r1_ordinal1(A,B) <=> r1_tarski(A,B) ) ) ), file(ordinal1,r1_ordinal1), [interesting(0.9),axiom,file(ordinal1,r1_ordinal1)]). fof(dt_k7_xcmplx_0,axiom,( $true ), file(xcmplx_0,k7_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k7_xcmplx_0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r0_r0,theorem,( k6_xcmplx_0(0,0) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r0_r0)]). fof(rqRealDiff__k6_xcmplx_0__r0_r1_rm1,theorem,( k6_xcmplx_0(0,1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm1_r1,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r0_r1,theorem,( k6_xcmplx_0(1,0) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r0_r1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),0) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1)]). fof(rqRealDiv__k7_xcmplx_0__r0_r1_r0,theorem,( k7_xcmplx_0(0,1) = 0 ), file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r0_r1_r0)]). fof(rqRealDiv__k7_xcmplx_0__r1_r1_r1,theorem,( k7_xcmplx_0(1,1) = 1 ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_r1_r1)]). fof(rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,theorem,( k7_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(1) ), file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1)]). fof(rqRealNeg__k4_xcmplx_0__r0_r0,theorem,( k4_xcmplx_0(0) = 0 ), file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r0_r0)]). fof(e4_5_1__card_4,plain,( c2_5_1__card_4 != k1_xboole_0 ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_zfmisc_1,dt_k5_ordinal2,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_card_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc1_subset_1,fc3_xreal_0,fc5_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_subset_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_subset,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_ordinal1,cc2_card_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,fc2_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc3_ordinal1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t7_boole,t8_boole,commutativity_k1_nat_1,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_r2_wellord2,dt_k1_nat_1,dt_k1_xboole_0,dt_c1_5_1__card_4,dt_c2_5_1__card_4,fc2_finseq_1,fc2_ordinal1,fc6_membered,t6_boole,spc1_numerals,spc1_boole,e2_5_1__card_4,t46_card_1]), [interesting(0.65),file(card_4,e4_5_1__card_4),[file(card_4,e4_5_1__card_4)]]). fof(t10_ordinal3,theorem,( ! [A] : ( v3_ordinal1(A) => ( A != k1_xboole_0 => r2_hidden(k1_xboole_0,A) ) ) ), file(ordinal3,t10_ordinal3), [interesting(0.9),axiom,file(ordinal3,t10_ordinal3)]). fof(e5_5_1__card_4,plain,( r2_hidden(k1_xboole_0,c2_5_1__card_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[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_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,existence_m1_subset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_membered,cc3_ordinal1,cc4_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc3_ordinal1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t8_boole,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c2_5_1__card_4,cc1_ordinal1,fc2_finseq_1,fc2_ordinal1,fc6_membered,t1_subset,t6_boole,t7_boole,e4_5_1__card_4,t10_ordinal3]), [interesting(0.65),file(card_4,e5_5_1__card_4),[file(card_4,e5_5_1__card_4)]]). fof(d5_card_1,definition,( ! [A,B] : ( v1_card_1(B) => ( B = k1_card_1(A) <=> r2_wellord2(A,B) ) ) ), file(card_1,d5_card_1), [interesting(0.9),axiom,file(card_1,d5_card_1)]). fof(d5_ordinal2,definition,( ! [A] : ( A = k5_ordinal2 <=> ( r2_hidden(k1_xboole_0,A) & v4_ordinal1(A) & v3_ordinal1(A) & ! [B] : ( v3_ordinal1(B) => ( ( r2_hidden(k1_xboole_0,B) & v4_ordinal1(B) ) => r1_tarski(A,B) ) ) ) ) ), file(ordinal2,d5_ordinal2), [interesting(0.9),axiom,file(ordinal2,d5_ordinal2)]). fof(e2_5_1_1__card_4,plain, ( r1_ordinal1(k5_ordinal2,c2_5_1__card_4) & r2_hidden(k1_nat_1(c1_5_1__card_4,1),k5_ordinal2) & k1_card_1(k1_nat_1(c1_5_1__card_4,1)) = k1_nat_1(c1_5_1__card_4,1) ), inference(mizar_by,[status(thm),assumptions([e1_5_1_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc3_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_xreal_0,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc1_subset_1,fc2_card_1,fc2_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc1_ordinal2,rc1_subset_1,rc2_card_1,rc2_funct_1,rc2_ordinal1,rc2_subset_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc4_finset_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t4_subset,t5_subset,t8_boole,commutativity_k1_nat_1,reflexivity_r1_ordinal1,connectedness_r1_ordinal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_r1_ordinal1,redefinition_r2_wellord2,dt_k1_card_1,dt_k1_nat_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_c1_5_1__card_4,dt_c2_5_1__card_4,cc1_card_1,cc1_ordinal1,fc1_ordinal2,fc2_finseq_1,fc2_ordinal1,fc5_membered,fc6_membered,rc1_card_1,t1_subset,t3_subset,t6_boole,t7_boole,spc1_numerals,spc1_boole,e1_5_1_1__card_4,e5_5_1__card_4,d5_card_1,d5_ordinal2]), [interesting(0.5),file(card_4,e2_5_1_1__card_4),[file(card_4,e2_5_1_1__card_4)]]). fof(t21_card_1,theorem,( ! [A,B] : ( r2_wellord2(A,B) <=> k1_card_1(A) = k1_card_1(B) ) ), file(card_1,t21_card_1), [interesting(0.9),axiom,file(card_1,t21_card_1)]). fof(t27_card_1,theorem,( ! [A,B] : ( r1_tarski(A,B) => r1_tarski(k1_card_1(A),k1_card_1(B)) ) ), file(card_1,t27_card_1), [interesting(0.9),axiom,file(card_1,t27_card_1)]). fof(t84_card_1,theorem,( k1_card_1(k5_ordinal2) = k5_ordinal2 ), file(card_1,t84_card_1), [interesting(0.9),axiom,file(card_1,t84_card_1)]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,theorem,( k7_xcmplx_0(1,k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1)]). fof(rqRealNeg__k4_xcmplx_0__rm1_r1,theorem,( k4_xcmplx_0(k4_xcmplx_0(1)) = 1 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r1_r1_r0,theorem,( k6_xcmplx_0(1,1) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r1_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0)]). fof(e3_5_1_1__card_4,plain, ( r1_tarski(k1_card_1(k5_ordinal2),k1_card_1(c2_5_1__card_4)) & k1_card_1(c2_5_1__card_4) = k1_card_1(k1_nat_1(c1_5_1__card_4,1)) & r2_hidden(k1_card_1(k1_nat_1(c1_5_1__card_4,1)),k1_card_1(k5_ordinal2)) ), inference(mizar_by,[status(thm),assumptions([e1_5_1_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_xreal_0,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_card_1,fc2_finseq_1,fc2_ordinal1,fc30_xreal_0,fc3_xreal_0,fc5_xreal_0,fc6_membered,fc6_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_xreal_0,rc2_card_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_card_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_ordinal1,cc20_membered,cc2_card_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc1_subset_1,fc2_membered,rc1_card_1,rc1_membered,rc1_ordinal1,rc1_subset_1,rc2_subset_1,rc3_ordinal1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t4_arithm,t4_subset,t5_arithm,t5_subset,t6_arithm,t6_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_ordinal1,connectedness_r1_ordinal1,reflexivity_r1_tarski,antisymmetry_r2_hidden,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_r1_ordinal1,redefinition_r2_wellord2,dt_k1_card_1,dt_k1_nat_1,dt_k4_xcmplx_0,dt_k5_ordinal2,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_5_1__card_4,dt_c2_5_1__card_4,fc1_ordinal2,fc5_membered,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqRealNeg__k4_xcmplx_0__r0_r0,t1_subset,t3_subset,t7_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e2_5_1_1__card_4,e2_5_1__card_4,t21_card_1,t27_card_1,t84_card_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0]), [interesting(0.5),file(card_4,e3_5_1_1__card_4),[file(card_4,e3_5_1_1__card_4)]]). fof(t14_card_1,theorem,( ! [A] : ( v1_card_1(A) => ! [B] : ( v1_card_1(B) => ( r2_hidden(A,B) <=> ~ r1_tarski(B,A) ) ) ) ), file(card_1,t14_card_1), [interesting(0.9),axiom,file(card_1,t14_card_1)]). fof(e4_5_1_1__card_4,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([e1_5_1_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc2_card_1,fc2_finseq_1,fc2_ordinal1,fc3_xreal_0,fc6_membered,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finset_1,rc1_xreal_0,rc2_card_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_ordinal1,cc20_membered,cc2_card_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc1_subset_1,fc2_membered,rc1_membered,rc1_ordinal1,rc1_subset_1,rc2_subset_1,rc3_ordinal1,t2_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k1_nat_1,dt_k1_card_1,dt_k1_nat_1,dt_k5_ordinal2,dt_c1_5_1__card_4,dt_c2_5_1__card_4,cc1_card_1,fc1_ordinal2,fc5_membered,rc1_card_1,t1_subset,t3_subset,t7_boole,spc1_numerals,spc1_boole,e3_5_1_1__card_4,t14_card_1]), [interesting(0.5),file(card_4,e4_5_1_1__card_4),[file(card_4,e4_5_1_1__card_4)]]). fof(i2_5_1_1__card_4,theorem,( $true ), introduced(tautology,[file(card_4,i2_5_1_1__card_4)]), [interesting(0.5),trivial,file(card_4,i2_5_1_1__card_4)]). fof(i1_5_1_1__card_4,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([e1_5_1_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[e4_5_1_1__card_4,i2_5_1_1__card_4]), [interesting(0.5),file(card_4,i1_5_1_1__card_4),[file(card_4,i1_5_1_1__card_4)]]). fof(e6_5_1__card_4,plain,( ~ v4_ordinal1(c2_5_1__card_4) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4]),discharge_asm(discharge,[e1_5_1_1__card_4])],[e1_5_1_1__card_4,i1_5_1_1__card_4]), [interesting(0.65),file(card_4,e6_5_1__card_4),[file(card_4,e6_5_1__card_4)]]). fof(t42_ordinal1,theorem,( ! [A] : ( v3_ordinal1(A) => ( ~ ( ~ v4_ordinal1(A) & ! [B] : ( v3_ordinal1(B) => A != k1_ordinal1(B) ) ) & ~ ( ? [B] : ( v3_ordinal1(B) & A = k1_ordinal1(B) ) & v4_ordinal1(A) ) ) ) ), file(ordinal1,t42_ordinal1), [interesting(0.9),axiom,file(ordinal1,t42_ordinal1)]). fof(e7_5_1__card_4,plain,( ? [A] : ( v3_ordinal1(A) & c2_5_1__card_4 = k1_ordinal1(A) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[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_xreal_0,cc2_xreal_0,cc7_xreal_0,rc1_xreal_0,existence_m1_subset_1,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_membered,cc2_membered,cc3_membered,cc4_membered,fc2_finseq_1,fc2_ordinal1,fc6_membered,rc1_finset_1,rc1_membered,t1_subset,cc15_membered,cc1_finset_1,cc1_funct_1,cc2_ordinal1,cc3_ordinal1,rc1_ordinal1,rc1_ordinal2,rc3_ordinal1,t6_boole,t7_boole,t8_boole,dt_k1_ordinal1,dt_c2_5_1__card_4,cc1_ordinal1,fc1_ordinal1,fc3_ordinal1,e6_5_1__card_4,t42_ordinal1]), [interesting(0.65),file(card_4,e7_5_1__card_4),[file(card_4,e7_5_1__card_4)]]). fof(dt_c3_5_1__card_4,plain,( v3_ordinal1(c3_5_1__card_4) ), inference(consider,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[dh_c3_5_1__card_4,e7_5_1__card_4]), [interesting(0.65),file(card_4,c3_5_1__card_4),[file(card_4,c3_5_1__card_4)]]). 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(fc2_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_tarski(A)) ), file(subset_1,fc2_subset_1), [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]). fof(rqRealDiff__k6_xcmplx_0__r0_r2_rm2,theorem,( k6_xcmplx_0(0,2) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_r2_rm2)]). fof(rqRealDiff__k6_xcmplx_0__r0_rm2_r2,theorem,( k6_xcmplx_0(0,k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r0_rm2_r2)]). fof(rqRealDiff__k6_xcmplx_0__r1_r2_rm1,theorem,( k6_xcmplx_0(1,2) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_r2_rm1)]). fof(rqRealDiff__k6_xcmplx_0__r1_rm1_r2,theorem,( k6_xcmplx_0(1,k4_xcmplx_0(1)) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r1_rm1_r2)]). fof(rqRealDiff__k6_xcmplx_0__r2_r0_r2,theorem,( k6_xcmplx_0(2,0) = 2 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r0_r2)]). fof(rqRealDiff__k6_xcmplx_0__r2_r1_r1,theorem,( k6_xcmplx_0(2,1) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r1_r1)]). fof(rqRealDiff__k6_xcmplx_0__r2_r2_r0,theorem,( k6_xcmplx_0(2,2) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__r2_r2_r0)]). fof(rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) = 1 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1)]). fof(rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),0) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) = k4_xcmplx_0(1) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1)]). fof(rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,theorem,( k6_xcmplx_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) = 0 ), file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0)]). fof(rqRealNeg__k4_xcmplx_0__r2_rm2,theorem,( k4_xcmplx_0(2) = k4_xcmplx_0(2) ), file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r2_rm2)]). fof(rqRealNeg__k4_xcmplx_0__rm2_r2,theorem,( k4_xcmplx_0(k4_xcmplx_0(2)) = 2 ), file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__rm2_r2)]). fof(rqSucc__k1_ordinal1__r0_r1,theorem,( k1_ordinal1(0) = 1 ), file(arithm,rqSucc__k1_ordinal1__r0_r1), [interesting(0.9),axiom,file(arithm,rqSucc__k1_ordinal1__r0_r1)]). fof(rqSucc__k1_ordinal1__r1_r2,theorem,( k1_ordinal1(1) = 2 ), file(arithm,rqSucc__k1_ordinal1__r1_r2), [interesting(0.9),axiom,file(arithm,rqSucc__k1_ordinal1__r1_r2)]). fof(t3_boole,theorem,( ! [A] : k4_xboole_0(A,k1_xboole_0) = A ), file(boole,t3_boole), [interesting(0.9),axiom,file(boole,t3_boole)]). fof(t4_boole,theorem,( ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ), file(boole,t4_boole), [interesting(0.9),axiom,file(boole,t4_boole)]). fof(spc2_numerals,theorem, ( v2_xreal_0(2) & m2_subset_1(2,k1_numbers,k5_numbers) & m1_subset_1(2,k5_numbers) & m1_subset_1(2,k1_numbers) ), file(numerals,spc2_numerals), [interesting(0.9),axiom,file(numerals,spc2_numerals)]). fof(spc2_boole,theorem,( ~ v1_xboole_0(2) ), file(boole,spc2_boole), [interesting(0.9),axiom,file(boole,spc2_boole)]). fof(t52_card_1,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => k1_ordinal1(A) = k1_nat_1(A,1) ) ), file(card_1,t52_card_1), [interesting(0.9),axiom,file(card_1,t52_card_1)]). fof(e3_5_1__card_4,plain, ( k1_nat_1(c1_5_1__card_4,1) = k1_ordinal1(c1_5_1__card_4) & k1_ordinal1(c1_5_1__card_4) != k1_xboole_0 & r2_wellord2(k1_nat_1(c1_5_1__card_4,1),c2_5_1__card_4) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc1_card_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc3_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_ordinal1,cc20_membered,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_subset_1,fc3_ordinal1,fc5_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc1_subset_1,rc2_funct_1,rc2_ordinal1,rc2_subset_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc4_finset_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t7_boole,t8_boole,commutativity_k1_nat_1,symmetry_r2_wellord2,reflexivity_r2_wellord2,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,redefinition_r2_wellord2,dt_k1_nat_1,dt_k1_numbers,dt_k1_ordinal1,dt_k1_xboole_0,dt_k5_numbers,dt_m2_subset_1,dt_c1_5_1__card_4,dt_c2_5_1__card_4,fc1_ordinal1,fc2_finseq_1,fc2_membered,fc2_ordinal1,fc6_membered,t6_boole,spc1_numerals,spc1_boole,e2_5_1__card_4,t52_card_1]), [interesting(0.65),file(card_4,e3_5_1__card_4),[file(card_4,e3_5_1__card_4)]]). fof(e8_5_1__card_4,plain,( c2_5_1__card_4 = k1_ordinal1(c3_5_1__card_4) ), inference(consider,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[dh_c3_5_1__card_4,e7_5_1__card_4]), [interesting(0.65),file(card_4,e8_5_1__card_4),[file(card_4,e8_5_1__card_4)]]). fof(t10_ordinal1,theorem,( ! [A] : r2_hidden(A,k1_ordinal1(A)) ), file(ordinal1,t10_ordinal1), [interesting(0.9),axiom,file(ordinal1,t10_ordinal1)]). fof(e9_5_1__card_4,plain, ( r2_hidden(c3_5_1__card_4,c2_5_1__card_4) & r2_hidden(c1_5_1__card_4,k1_nat_1(c1_5_1__card_4,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[reflexivity_r1_tarski,dt_k1_zfmisc_1,dt_k5_ordinal2,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_card_1,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_membered,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,cc9_membered,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_ordinal2,fc1_subset_1,fc1_xreal_0,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc30_xreal_0,fc3_xreal_0,fc5_membered,fc5_xreal_0,fc6_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_subset_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_subset_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t3_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc1_ordinal1,cc2_card_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,fc2_membered,fc3_ordinal1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc2_funct_1,rc2_ordinal1,rc3_finseq_1,rc3_funct_1,rc3_ordinal1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t4_arithm,t5_arithm,t6_arithm,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,antisymmetry_r2_hidden,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_r2_wellord2,dt_k1_nat_1,dt_k1_ordinal1,dt_k1_xboole_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_5_1__card_4,dt_c2_5_1__card_4,dt_c3_5_1__card_4,fc1_ordinal1,fc2_finseq_1,fc2_ordinal1,fc6_membered,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiv__k7_xcmplx_0__r0_r1_r0,rqRealDiv__k7_xcmplx_0__r1_r1_r1,rqRealDiv__k7_xcmplx_0__rm1_r1_rm1,rqSucc__k1_ordinal1__r0_r1,t1_subset,t6_boole,t7_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e3_5_1__card_4,e8_5_1__card_4,t10_ordinal1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.65),file(card_4,e9_5_1__card_4),[file(card_4,e9_5_1__card_4)]]). fof(t5_card_4,theorem,( ! [A] : ( v3_ordinal1(A) => k4_xboole_0(k1_ordinal1(A),k1_tarski(A)) = A ) ), file(card_4,t5_card_4), [interesting(0.9),axiom,file(card_4,t5_card_4)]). fof(t61_card_1,theorem,( ! [A,B,C,D] : ( ( r2_wellord2(A,B) & r2_hidden(C,A) & r2_hidden(D,B) ) => r2_wellord2(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(B,k1_tarski(D))) ) ), file(card_1,t61_card_1), [interesting(0.9),axiom,file(card_1,t61_card_1)]). fof(rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,theorem,( k6_xcmplx_0(k4_xcmplx_0(1),1) = k4_xcmplx_0(2) ), file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2), [interesting(0.9),axiom,file(arithm,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2)]). fof(e10_5_1__card_4,plain, ( r2_wellord2(k4_xboole_0(c2_5_1__card_4,k1_tarski(c3_5_1__card_4)),k4_xboole_0(k1_nat_1(c1_5_1__card_4,1),k6_domain_1(k5_numbers,c1_5_1__card_4))) & k4_xboole_0(c2_5_1__card_4,k1_tarski(c3_5_1__card_4)) = c3_5_1__card_4 & k4_xboole_0(k1_nat_1(c1_5_1__card_4,1),k6_domain_1(k5_numbers,c1_5_1__card_4)) = c1_5_1__card_4 ), inference(mizar_by,[status(thm),assumptions([dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[reflexivity_r1_tarski,cc1_card_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc10_xreal_0,fc11_membered,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_xreal_0,fc20_xreal_0,fc3_xreal_0,fc5_xreal_0,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_xreal_0,rc1_card_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc12_finset_1,fc1_ordinal2,fc1_subset_1,fc2_membered,fc37_membered,fc38_membered,fc39_membered,fc40_membered,fc41_membered,fc5_membered,fc7_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc1_subset_1,rc2_funct_1,rc2_ordinal1,rc2_subset_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc4_finset_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t4_subset,t5_subset,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,antisymmetry_r2_hidden,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_domain_1,redefinition_r2_wellord2,dt_k1_nat_1,dt_k1_ordinal1,dt_k1_tarski,dt_k1_xboole_0,dt_k4_xboole_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_domain_1,dt_k6_xcmplx_0,dt_c1_5_1__card_4,dt_c2_5_1__card_4,dt_c3_5_1__card_4,cc1_ordinal1,fc1_finset_1,fc1_ordinal1,fc2_finseq_1,fc2_ordinal1,fc2_subset_1,fc3_ordinal1,fc6_membered,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,rqSucc__k1_ordinal1__r0_r1,rqSucc__k1_ordinal1__r1_r2,t1_subset,t3_boole,t4_boole,t6_boole,t7_boole,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e9_5_1__card_4,e3_5_1__card_4,e8_5_1__card_4,t5_card_4,t61_card_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.65),file(card_4,e10_5_1__card_4),[file(card_4,e10_5_1__card_4)]]). fof(e11_5_1__card_4,plain,( c2_5_1__card_4 = k1_nat_1(c1_5_1__card_4,1) ), inference(mizar_by,[status(thm),assumptions([e1_5_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,cc1_card_1,cc1_xreal_0,cc2_xreal_0,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_membered,fc10_xreal_0,fc11_membered,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_xreal_0,fc20_xreal_0,fc3_xreal_0,fc5_xreal_0,fc7_xreal_0,fc8_membered,fc8_xreal_0,fc9_membered,fc9_xreal_0,rc1_card_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,t1_subset,t4_subset,t5_subset,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_subset_1,cc10_membered,cc11_membered,cc12_membered,cc13_membered,cc14_membered,cc15_membered,cc16_membered,cc17_membered,cc18_membered,cc19_membered,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_membered,cc20_membered,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_membered,cc2_ordinal1,cc3_card_1,cc3_membered,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc12_finset_1,fc1_ordinal2,fc1_subset_1,fc2_membered,fc37_membered,fc38_membered,fc39_membered,fc40_membered,fc41_membered,fc5_membered,fc7_membered,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_membered,rc1_ordinal1,rc1_subset_1,rc2_funct_1,rc2_ordinal1,rc2_subset_1,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_ordinal1,rc4_finset_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r0_r2_r2,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_rm2_rm2,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqRealAdd__k2_xcmplx_0__r1_r1_r2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r1_rm2_rm1,rqRealAdd__k2_xcmplx_0__r2_r0_r2,rqRealAdd__k2_xcmplx_0__r2_rm1_r1,rqRealAdd__k2_xcmplx_0__r2_rm2_r0,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealAdd__k2_xcmplx_0__rm1_r2_r1,rqRealAdd__k2_xcmplx_0__rm1_rm1_rm2,rqRealAdd__k2_xcmplx_0__rm2_r0_rm2,rqRealAdd__k2_xcmplx_0__rm2_r1_rm1,rqRealAdd__k2_xcmplx_0__rm2_r2_r0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_k6_domain_1,redefinition_r2_wellord2,dt_k1_nat_1,dt_k1_ordinal1,dt_k1_tarski,dt_k1_xboole_0,dt_k4_xboole_0,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_domain_1,dt_k6_xcmplx_0,dt_c1_5_1__card_4,dt_c2_5_1__card_4,dt_c3_5_1__card_4,cc1_ordinal1,fc1_finset_1,fc1_ordinal1,fc2_finseq_1,fc2_ordinal1,fc2_subset_1,fc3_ordinal1,fc6_membered,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r2_rm2,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r0_rm2_r2,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm2_r1,rqRealDiff__k6_xcmplx_0__rm2_r0_rm2,rqRealDiff__k6_xcmplx_0__rm2_rm1_rm1,rqRealDiff__k6_xcmplx_0__rm2_rm2_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,rqSucc__k1_ordinal1__r0_r1,rqSucc__k1_ordinal1__r1_r2,t3_boole,t4_boole,t6_boole,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e10_5_1__card_4,e1_5_1__card_4,e3_5_1__card_4,e8_5_1__card_4,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r0_r1_rm1]), [interesting(0.65),file(card_4,e11_5_1__card_4),[file(card_4,e11_5_1__card_4)]]). fof(i5_5_1__card_4,theorem,( $true ), introduced(tautology,[file(card_4,i5_5_1__card_4)]), [interesting(0.65),trivial,file(card_4,i5_5_1__card_4)]). fof(i4_5_1__card_4,plain,( c2_5_1__card_4 = k1_nat_1(c1_5_1__card_4,1) ), inference(conclusion,[status(thm),assumptions([e1_5_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4,e2_5_1__card_4])],[e11_5_1__card_4,i5_5_1__card_4]), [interesting(0.65),file(card_4,i4_5_1__card_4),[file(card_4,i4_5_1__card_4)]]). fof(i3_5_1__card_4,plain, ( r2_wellord2(c2_5_1__card_4,k1_nat_1(c1_5_1__card_4,1)) => c2_5_1__card_4 = k1_nat_1(c1_5_1__card_4,1) ), inference(discharge_asm,[status(thm),assumptions([e1_5_1__card_4,dt_c1_5_1__card_4,dt_c2_5_1__card_4]),discharge_asm(discharge,[e2_5_1__card_4])],[e2_5_1__card_4,i4_5_1__card_4]), [interesting(0.65),file(card_4,i3_5_1__card_4),[file(card_4,i3_5_1__card_4)]]). fof(i3_5_1_tmp__card_4,plain, ( v3_ordinal1(c2_5_1__card_4) => ( r2_wellord2(c2_5_1__card_4,k1_nat_1(c1_5_1__card_4,1)) => c2_5_1__card_4 = k1_nat_1(c1_5_1__card_4,1) ) ), inference(discharge_asm,[status(thm),assumptions([e1_5_1__card_4,dt_c1_5_1__card_4]),discharge_asm(discharge,[dt_c2_5_1__card_4])],[dt_c2_5_1__card_4,i3_5_1__card_4]), [interesting(0.65),i2_5_1__card_4]). fof(i2_5_1__card_4,plain,( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,k1_nat_1(c1_5_1__card_4,1)) => A = k1_nat_1(c1_5_1__card_4,1) ) ) ), inference(let,[status(thm),assumptions([e1_5_1__card_4,dt_c1_5_1__card_4])],[i3_5_1_tmp__card_4,dh_c2_5_1__card_4]), [interesting(0.65),file(card_4,i2_5_1__card_4),[file(card_4,i2_5_1__card_4)]]). fof(i1_5_1__card_4,plain, ( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5_1__card_4) => A = c1_5_1__card_4 ) ) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,k1_nat_1(c1_5_1__card_4,1)) => A = k1_nat_1(c1_5_1__card_4,1) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5_1__card_4]),discharge_asm(discharge,[e1_5_1__card_4])],[e1_5_1__card_4,i2_5_1__card_4]), [interesting(0.65),file(card_4,i1_5_1__card_4),[file(card_4,i1_5_1__card_4)]]). fof(i1_5_1_tmp__card_4,plain, ( m2_subset_1(c1_5_1__card_4,k1_numbers,k5_numbers) => ( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5_1__card_4) => A = c1_5_1__card_4 ) ) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,k1_nat_1(c1_5_1__card_4,1)) => A = k1_nat_1(c1_5_1__card_4,1) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5_1__card_4])],[dt_c1_5_1__card_4,i1_5_1__card_4]), [interesting(0.8),e2_5__card_4]). fof(e2_5__card_4,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ! [B] : ( v3_ordinal1(B) => ( r2_wellord2(B,A) => B = A ) ) => ! [B] : ( v3_ordinal1(B) => ( r2_wellord2(B,k1_nat_1(A,1)) => B = k1_nat_1(A,1) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_5_1_tmp__card_4,dh_c1_5_1__card_4]), [interesting(0.8),file(card_4,e2_5__card_4),[file(card_4,e2_5__card_4)]]). fof(e3_5__card_4,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v3_ordinal1(B) => ( r2_wellord2(B,A) => B = A ) ) ) ), inference(mizar_from,[status(thm),assumptions([])],[cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_card_1,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc3_xreal_0,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc3_xreal_0,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_card_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,commutativity_k2_xcmplx_0,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_card_1,cc2_ordinal1,cc3_card_1,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_subset_1,fc5_membered,rc1_ordinal1,rc1_subset_1,rc2_subset_1,rc3_ordinal1,commutativity_k1_nat_1,symmetry_r2_wellord2,reflexivity_r2_wellord2,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,redefinition_r2_wellord2,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_ordinal1,fc2_membered,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,s1_nat_1__e3_5__card_4,e1_5__card_4,e2_5__card_4]), [interesting(0.8),file(card_4,e3_5__card_4),[file(card_4,e3_5__card_4)]]). fof(e4_5__card_4,plain, ( r2_wellord2(c2_5__card_4,c1_5__card_4) => c2_5__card_4 = c1_5__card_4 ), inference(mizar_by,[status(thm),assumptions([dt_c1_5__card_4,dt_c2_5__card_4])],[cc1_finseq_1,cc2_funct_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc2_ordinal1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc12_membered,cc13_membered,cc14_membered,cc18_membered,cc19_membered,cc1_card_1,cc1_membered,cc1_xreal_0,cc20_membered,cc2_finset_1,cc2_membered,cc2_xreal_0,cc3_membered,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc2_finseq_1,fc2_ordinal1,fc6_membered,rc1_card_1,rc1_finset_1,rc1_membered,rc1_xreal_0,rc2_card_1,rc3_finset_1,rc4_finset_1,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc10_membered,cc11_membered,cc15_membered,cc16_membered,cc17_membered,cc1_finset_1,cc1_funct_1,cc2_card_1,cc2_ordinal1,cc3_card_1,cc3_ordinal1,cc4_membered,cc6_membered,cc9_membered,fc1_ordinal2,fc1_subset_1,fc5_membered,rc1_ordinal1,rc1_subset_1,rc2_subset_1,rc3_ordinal1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,symmetry_r2_wellord2,reflexivity_r2_wellord2,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,redefinition_r2_wellord2,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_5__card_4,dt_c2_5__card_4,cc1_ordinal1,fc2_membered,e3_5__card_4]), [interesting(0.8),file(card_4,e4_5__card_4),[file(card_4,e4_5__card_4)]]). fof(i3_5__card_4,theorem,( $true ), introduced(tautology,[file(card_4,i3_5__card_4)]), [interesting(0.8),trivial,file(card_4,i3_5__card_4)]). fof(i2_5__card_4,plain, ( r2_wellord2(c2_5__card_4,c1_5__card_4) => c2_5__card_4 = c1_5__card_4 ), inference(conclusion,[status(thm),assumptions([dt_c1_5__card_4,dt_c2_5__card_4])],[e4_5__card_4,i3_5__card_4]), [interesting(0.8),file(card_4,i2_5__card_4),[file(card_4,i2_5__card_4)]]). fof(i2_5_tmp__card_4,plain, ( v3_ordinal1(c2_5__card_4) => ( r2_wellord2(c2_5__card_4,c1_5__card_4) => c2_5__card_4 = c1_5__card_4 ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_5__card_4]),discharge_asm(discharge,[dt_c2_5__card_4])],[dt_c2_5__card_4,i2_5__card_4]), [interesting(0.8),i1_5__card_4]). fof(i1_5__card_4,plain,( ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5__card_4) => A = c1_5__card_4 ) ) ), inference(let,[status(thm),assumptions([dt_c1_5__card_4])],[i2_5_tmp__card_4,dh_c2_5__card_4]), [interesting(0.8),file(card_4,i1_5__card_4),[file(card_4,i1_5__card_4)]]). fof(i1_5_tmp__card_4,plain, ( m2_subset_1(c1_5__card_4,k1_numbers,k5_numbers) => ! [A] : ( v3_ordinal1(A) => ( r2_wellord2(A,c1_5__card_4) => A = c1_5__card_4 ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__card_4])],[dt_c1_5__card_4,i1_5__card_4]), [interesting(1),t6_card_4]). fof(t6_card_4,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( v3_ordinal1(B) => ( r2_wellord2(B,A) => B = A ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_5_tmp__card_4,dh_c1_5__card_4]), [interesting(1),file(card_4,t6_card_4),[file(card_4,t6_card_4)]]).