% Mizar ND problem: t5_graph_2,graph_2,246,53 fof(dh_c1_7__graph_2,definition, ( ( m2_subset_1(c1_7__graph_2,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,A)),B) = k1_nat_1(A,B) ) ) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,E) & r1_xreal_0(E,C) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(C,D)),E) = k1_nat_1(D,E) ) ) ) ) ), introduced(definition,[new_symbol(c1_7__graph_2),file(graph_2,c1_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c1_7__graph_2)]). fof(dh_c2_7__graph_2,definition, ( ( m2_subset_1(c2_7__graph_2,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) = k1_nat_1(c2_7__graph_2,A) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,B)),C) = k1_nat_1(B,C) ) ) ) ), introduced(definition,[new_symbol(c2_7__graph_2),file(graph_2,c2_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c2_7__graph_2)]). fof(e1_7__graph_2,assumption,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) != k1_nat_1(c2_7__graph_2,A) ) ), introduced(assumption,[file(graph_2,e1_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,e1_7__graph_2)]). fof(e1_7_4_1__graph_2,assumption,( ~ r1_xreal_0(c8_7__graph_2,c4_7__graph_2) ), introduced(assumption,[file(graph_2,e1_7_4_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,e1_7_4_1__graph_2)]). 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(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k2_zfmisc_1,axiom,( $true ), file(zfmisc_1,k2_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]). 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_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(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(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(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(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(rc2_finset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_ordinal1(B) & v2_ordinal1(B) & v3_ordinal1(B) & v4_ordinal2(B) & v1_finset_1(B) ) ), file(finset_1,rc2_finset_1), [interesting(0.9),axiom,file(finset_1,rc2_finset_1)]). fof(rc2_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k1_zfmisc_1(k1_numbers)) & ~ v1_xboole_0(A) & v3_ordinal1(A) ) ), file(nat_1,rc2_nat_1), [interesting(0.9),axiom,file(nat_1,rc2_nat_1)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc3_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_nat_1,theorem,( ? [A] : ( m1_subset_1(A,k5_numbers) & ~ v1_xboole_0(A) & v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc3_nat_1), [interesting(0.9),axiom,file(nat_1,rc3_nat_1)]). fof(rc4_finseq_1,theorem,( ! [A] : ? [B] : ( m1_finseq_1(B,A) & v1_xboole_0(B) & v1_relat_1(B) & v1_funct_1(B) & v2_funct_1(B) & v1_finset_1(B) & v1_finseq_1(B) ) ), file(finseq_1,rc4_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc4_finseq_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(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(existence_m1_finseq_1,axiom,( ! [A] : ? [B] : m1_finseq_1(B,A) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(existence_m1_relset_1,axiom,( ! [A,B] : ? [C] : m1_relset_1(C,A,B) ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(existence_m1_subset_1,axiom,( ! [A] : ? [B] : m1_subset_1(B,A) ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(existence_m2_relset_1,axiom,( ! [A,B] : ? [C] : m2_relset_1(C,A,B) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(existence_m2_subset_1,axiom,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ? [C] : m2_subset_1(C,A,B) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(redefinition_m2_relset_1,definition,( ! [A,B,C] : ( m2_relset_1(C,A,B) <=> m1_relset_1(C,A,B) ) ), file(relset_1,m2_relset_1), [interesting(0.9),axiom,file(relset_1,m2_relset_1)]). fof(redefinition_m2_subset_1,definition,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) & m1_subset_1(B,k1_zfmisc_1(A)) ) => ! [C] : ( m2_subset_1(C,A,B) <=> m1_subset_1(C,B) ) ) ), file(subset_1,m2_subset_1), [interesting(0.9),axiom,file(subset_1,m2_subset_1)]). fof(dt_k1_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(dt_k1_finseq_1,axiom,( $true ), file(finseq_1,k1_finseq_1), [interesting(0.9),axiom,file(finseq_1,k1_finseq_1)]). fof(dt_k1_numbers,axiom,( $true ), file(numbers,k1_numbers), [interesting(0.9),axiom,file(numbers,k1_numbers)]). 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_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k2_xcmplx_0,axiom,( $true ), file(xcmplx_0,k2_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k2_xcmplx_0)]). fof(dt_k5_ordinal2,axiom,( $true ), file(ordinal2,k5_ordinal2), [interesting(0.9),axiom,file(ordinal2,k5_ordinal2)]). fof(dt_m1_finseq_1,axiom,( ! [A,B] : ( m1_finseq_1(B,A) => ( v1_relat_1(B) & v1_funct_1(B) & v1_finseq_1(B) ) ) ), file(finseq_1,m1_finseq_1), [interesting(0.9),axiom,file(finseq_1,m1_finseq_1)]). fof(dt_m1_relset_1,axiom,( $true ), file(relset_1,m1_relset_1), [interesting(0.9),axiom,file(relset_1,m1_relset_1)]). fof(dt_m1_subset_1,axiom,( $true ), file(subset_1,m1_subset_1), [interesting(0.9),axiom,file(subset_1,m1_subset_1)]). fof(dt_m2_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(cc1_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ( v1_relat_1(A) & v1_funct_1(A) & v1_finset_1(A) ) ) ), file(finseq_1,cc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,cc1_finseq_1)]). fof(cc1_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(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(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_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_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(cc2_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_int_1(A) ) ) ), file(int_1,cc2_int_1), [interesting(0.9),axiom,file(int_1,cc2_int_1)]). fof(cc2_nat_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_ordinal1(A) & v2_ordinal1(A) & v3_ordinal1(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc2_nat_1), [interesting(0.9),axiom,file(nat_1,cc2_nat_1)]). fof(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_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_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v2_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc3_xreal_0)]). fof(cc4_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(int_1,cc4_int_1), [interesting(0.9),axiom,file(int_1,cc4_int_1)]). fof(cc4_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v2_xreal_0(A) ) ) ), file(xreal_0,cc4_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc4_xreal_0)]). fof(cc5_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & v3_xreal_0(A) ) => ( ~ v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) ) ), file(xreal_0,cc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc5_xreal_0)]). fof(cc6_xreal_0,theorem,( ! [A] : ( ( ~ v1_xboole_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & v3_xreal_0(A) ) ) ), file(xreal_0,cc6_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc6_xreal_0)]). fof(cc7_xreal_0,theorem,( ! [A] : ( ( v1_xboole_0(A) & v1_xreal_0(A) ) => ( v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(xreal_0,cc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc7_xreal_0)]). fof(cc8_xreal_0,theorem,( ! [A] : ( ( v1_xreal_0(A) & ~ v2_xreal_0(A) & ~ v3_xreal_0(A) ) => ( v1_xboole_0(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ) ), file(xreal_0,cc8_xreal_0), [interesting(0.9),axiom,file(xreal_0,cc8_xreal_0)]). fof(fc10_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v2_xreal_0(k2_xcmplx_0(B,A)) & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(xreal_0,fc10_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc10_xreal_0)]). fof(fc11_finseq_1,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_finset_1(A) ) => v1_finset_1(k2_relat_1(A)) ) ), file(finseq_1,fc11_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc11_finseq_1)]). 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(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_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(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_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_finset_1(k1_finseq_1(A)) ) ), file(finseq_1,fc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,fc1_finseq_1)]). fof(fc1_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v1_int_1(k2_xcmplx_0(A,B)) ) ) ), file(int_1,fc1_int_1), [interesting(0.9),axiom,file(int_1,fc1_int_1)]). fof(fc1_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => ( v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc1_nat_1), [interesting(0.9),axiom,file(nat_1,fc1_nat_1)]). fof(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_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(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(fc3_int_1,theorem,( ! [A] : ( v1_int_1(A) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc3_int_1), [interesting(0.9),axiom,file(int_1,fc3_int_1)]). fof(fc3_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v4_ordinal2(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(nat_1,fc3_nat_1), [interesting(0.9),axiom,file(nat_1,fc3_nat_1)]). fof(fc3_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc3_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc3_xreal_0)]). fof(fc4_int_1,theorem,( ! [A,B] : ( ( v1_int_1(A) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc4_int_1), [interesting(0.9),axiom,file(int_1,fc4_int_1)]). fof(fc4_nat_1,theorem,( ! [A,B] : ( ( v4_ordinal2(A) & ~ v1_xboole_0(B) & v4_ordinal2(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(B,A)) & v4_ordinal2(k2_xcmplx_0(B,A)) & v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) ) ) ), file(nat_1,fc4_nat_1), [interesting(0.9),axiom,file(nat_1,fc4_nat_1)]). fof(fc5_int_1,theorem,( ! [A] : ( m1_subset_1(A,k5_numbers) => ( v1_xcmplx_0(k4_xcmplx_0(A)) & v1_xreal_0(k4_xcmplx_0(A)) & ~ v2_xreal_0(k4_xcmplx_0(A)) & v1_int_1(k4_xcmplx_0(A)) ) ) ), file(int_1,fc5_int_1), [interesting(0.9),axiom,file(int_1,fc5_int_1)]). fof(fc5_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v1_xreal_0(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc5_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc5_xreal_0)]). fof(fc6_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(B,A)) & v1_xreal_0(k2_xcmplx_0(B,A)) & v1_int_1(k2_xcmplx_0(B,A)) ) ) ), file(int_1,fc6_int_1), [interesting(0.9),axiom,file(int_1,fc6_int_1)]). fof(fc7_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & ~ v3_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc7_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc7_xreal_0)]). fof(fc8_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & v1_int_1(B) ) => ( v1_xcmplx_0(k6_xcmplx_0(B,A)) & v1_xreal_0(k6_xcmplx_0(B,A)) & v1_int_1(k6_xcmplx_0(B,A)) ) ) ), file(int_1,fc8_int_1), [interesting(0.9),axiom,file(int_1,fc8_int_1)]). fof(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_int_1,theorem,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => ( v1_xcmplx_0(k6_xcmplx_0(A,B)) & v1_xreal_0(k6_xcmplx_0(A,B)) & v1_int_1(k6_xcmplx_0(A,B)) ) ) ), file(int_1,fc9_int_1), [interesting(0.9),axiom,file(int_1,fc9_int_1)]). fof(fc9_xreal_0,theorem,( ! [A,B] : ( ( v1_xreal_0(A) & v2_xreal_0(A) & v1_xreal_0(B) & ~ v3_xreal_0(B) ) => ( ~ v1_xboole_0(k2_xcmplx_0(A,B)) & v1_xcmplx_0(k2_xcmplx_0(A,B)) & v1_xreal_0(k2_xcmplx_0(A,B)) & v2_xreal_0(k2_xcmplx_0(A,B)) & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) ), file(xreal_0,fc9_xreal_0), [interesting(0.9),axiom,file(xreal_0,fc9_xreal_0)]). fof(rc1_finseq_1,theorem,( ? [A] : ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc1_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc1_finseq_1)]). fof(rc1_finset_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v1_finset_1(A) ) ), file(finset_1,rc1_finset_1), [interesting(0.9),axiom,file(finset_1,rc1_finset_1)]). fof(rc1_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(rc1_int_1,theorem,( ? [A] : ( m1_subset_1(A,k1_numbers) & v1_xcmplx_0(A) & v1_xreal_0(A) & v1_int_1(A) ) ), file(int_1,rc1_int_1), [interesting(0.9),axiom,file(int_1,rc1_int_1)]). fof(rc1_nat_1,theorem,( ? [A] : ( ~ v1_xboole_0(A) & v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(nat_1,rc1_nat_1), [interesting(0.9),axiom,file(nat_1,rc1_nat_1)]). fof(rc1_xreal_0,theorem,( ? [A] : ( v1_xcmplx_0(A) & v1_xreal_0(A) ) ), file(xreal_0,rc1_xreal_0), [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]). fof(rc2_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_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_int_1,theorem,( ? [A] : v1_int_1(A) ), file(int_1,rc2_int_1), [interesting(0.9),axiom,file(int_1,rc2_int_1)]). fof(rc2_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_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(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(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(t1_numerals,theorem,( m1_subset_1(0,k5_numbers) ), file(numerals,t1_numerals), [interesting(0.9),axiom,file(numerals,t1_numerals)]). fof(t1_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v2_xreal_0(A) ) => v2_xreal_0(B) ) ) ) ), file(real,t1_real), [interesting(0.9),axiom,file(real,t1_real)]). fof(t2_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( ( r1_xreal_0(A,B) & v3_xreal_0(B) ) => v3_xreal_0(A) ) ) ) ), file(real,t2_real), [interesting(0.9),axiom,file(real,t2_real)]). fof(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_real,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ~ ( r1_xreal_0(A,B) & ~ v3_xreal_0(A) & v3_xreal_0(B) ) ) ) ), file(real,t3_real), [interesting(0.9),axiom,file(real,t3_real)]). fof(t4_arithm,theorem,( ! [A] : ( v1_xcmplx_0(A) => k6_xcmplx_0(A,0) = A ) ), file(arithm,t4_arithm), [interesting(0.9),axiom,file(arithm,t4_arithm)]). fof(t4_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(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_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(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(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_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(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(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(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(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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(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(existence_m2_finseq_1,axiom,( ! [A] : ? [B] : m2_finseq_1(B,A) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(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(redefinition_k1_recdef_1,definition,( ! [A,B,C] : ( ( v1_funct_1(B) & m1_relset_1(B,A,k5_numbers) & m1_subset_1(C,A) ) => k1_recdef_1(A,B,C) = k1_funct_1(B,C) ) ), file(recdef_1,k1_recdef_1), [interesting(0.9),axiom,file(recdef_1,k1_recdef_1)]). fof(redefinition_k2_finseq_1,definition,( ! [A] : ( v4_ordinal2(A) => k2_finseq_1(A) = k1_finseq_1(A) ) ), file(finseq_1,k2_finseq_1), [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]). fof(redefinition_k3_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k3_finseq_1(A) = k1_card_1(A) ) ), file(finseq_1,k3_finseq_1), [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]). fof(redefinition_k4_finseq_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => k4_finseq_1(A) = k1_relat_1(A) ) ), file(finseq_1,k4_finseq_1), [interesting(0.9),axiom,file(finseq_1,k4_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_finseq_1,definition,( ! [A,B] : ( m2_finseq_1(B,A) <=> m1_finseq_1(B,A) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(dt_k14_finseq_1,axiom,( ! [A] : m2_finseq_1(k14_finseq_1(A),k5_numbers) ), file(finseq_1,k14_finseq_1), [interesting(0.9),axiom,file(finseq_1,k14_finseq_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_nat_1,axiom,( ! [A,B] : ( ( m1_subset_1(A,k5_numbers) & m1_subset_1(B,k5_numbers) ) => m2_subset_1(k1_nat_1(A,B),k1_numbers,k5_numbers) ) ), file(nat_1,k1_nat_1), [interesting(0.9),axiom,file(nat_1,k1_nat_1)]). fof(dt_k1_recdef_1,axiom,( ! [A,B,C] : ( ( v1_funct_1(B) & m1_relset_1(B,A,k5_numbers) & m1_subset_1(C,A) ) => m2_subset_1(k1_recdef_1(A,B,C),k1_numbers,k5_numbers) ) ), file(recdef_1,k1_recdef_1), [interesting(0.9),axiom,file(recdef_1,k1_recdef_1)]). fof(dt_k2_finseq_1,axiom,( ! [A] : ( v4_ordinal2(A) => m1_subset_1(k2_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k2_finseq_1), [interesting(0.9),axiom,file(finseq_1,k2_finseq_1)]). fof(dt_k2_relat_1,axiom,( $true ), file(relat_1,k2_relat_1), [interesting(0.9),axiom,file(relat_1,k2_relat_1)]). fof(dt_k3_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m2_subset_1(k3_finseq_1(A),k1_numbers,k5_numbers) ) ), file(finseq_1,k3_finseq_1), [interesting(0.9),axiom,file(finseq_1,k3_finseq_1)]). fof(dt_k4_finseq_1,axiom,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => m1_subset_1(k4_finseq_1(A),k1_zfmisc_1(k5_numbers)) ) ), file(finseq_1,k4_finseq_1), [interesting(0.9),axiom,file(finseq_1,k4_finseq_1)]). 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_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_k6_xcmplx_0,axiom,( $true ), file(xcmplx_0,k6_xcmplx_0), [interesting(0.9),axiom,file(xcmplx_0,k6_xcmplx_0)]). fof(dt_m2_finseq_1,axiom,( ! [A,B] : ( m2_finseq_1(B,A) => ( v1_funct_1(B) & v1_finseq_1(B) & m2_relset_1(B,k5_numbers,A) ) ) ), file(finseq_1,m2_finseq_1), [interesting(0.9),axiom,file(finseq_1,m2_finseq_1)]). fof(dt_c1_7__graph_2,assumption,( m2_subset_1(c1_7__graph_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(graph_2,c1_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c1_7__graph_2)]). fof(dt_c2_7__graph_2,assumption,( m2_subset_1(c2_7__graph_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(graph_2,c2_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c2_7__graph_2)]). fof(dh_c4_7__graph_2,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) != k1_nat_1(c2_7__graph_2,A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),B) != k1_nat_1(c2_7__graph_2,B) ) => r1_xreal_0(A,B) ) ) ) => ( m2_subset_1(c4_7__graph_2,k1_numbers,k5_numbers) & r1_xreal_0(1,c4_7__graph_2) & r1_xreal_0(c4_7__graph_2,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c4_7__graph_2) != k1_nat_1(c2_7__graph_2,c4_7__graph_2) & ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),C) != k1_nat_1(c2_7__graph_2,C) ) => r1_xreal_0(c4_7__graph_2,C) ) ) ) ), introduced(definition,[new_symbol(c4_7__graph_2),file(graph_2,c4_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c4_7__graph_2)]). 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(cc3_int_1,theorem,( ! [A] : ( v4_ordinal2(A) => v1_int_1(A) ) ), file(int_1,cc3_int_1), [interesting(0.9),axiom,file(int_1,cc3_int_1)]). fof(cc3_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ( v4_ordinal2(A) & v1_xcmplx_0(A) & v1_xreal_0(A) & ~ v3_xreal_0(A) ) ) ), file(nat_1,cc3_nat_1), [interesting(0.9),axiom,file(nat_1,cc3_nat_1)]). fof(spc1_numerals,theorem, ( v2_xreal_0(1) & m2_subset_1(1,k1_numbers,k5_numbers) & m1_subset_1(1,k5_numbers) & m1_subset_1(1,k1_numbers) ), file(numerals,spc1_numerals), [interesting(0.9),axiom,file(numerals,spc1_numerals)]). fof(spc1_boole,theorem,( ~ v1_xboole_0(1) ), file(boole,spc1_boole), [interesting(0.9),axiom,file(boole,spc1_boole)]). fof(t2_tarski,theorem,( ! [A,B] : ( ! [C] : ( r2_hidden(C,A) <=> r2_hidden(C,B) ) => A = B ) ), file(tarski,t2_tarski), [interesting(0.9),axiom,file(tarski,t2_tarski)]). fof(fraenkel_a_2_1_graph_2,definition,( ! [A,B,C] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_1_graph_2(B,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = D & r1_xreal_0(k1_nat_1(C,1),D) & r1_xreal_0(D,k1_nat_1(C,B)) ) ) ) ), file(graph_2,a_2_1_graph_2), [interesting(0.9),axiom,file(graph_2,a_2_1_graph_2)]). fof(s5_nat_1__e2_7__graph_2,theorem,( ! [A,B] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(1,C) & r1_xreal_0(C,A) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),C) != k1_nat_1(B,C) ) => ? [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) & r1_xreal_0(1,C) & r1_xreal_0(C,A) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),C) != k1_nat_1(B,C) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,A) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),D) != k1_nat_1(B,D) ) => r1_xreal_0(C,D) ) ) ) ) ) ), file(graph_2,s5_nat_1__e2_7__graph_2), [interesting(0.9),axiom,file(graph_2,s5_nat_1__e2_7__graph_2)]). fof(e2_7__graph_2,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) != k1_nat_1(c2_7__graph_2,A) & ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),B) != k1_nat_1(c2_7__graph_2,B) ) => r1_xreal_0(A,B) ) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc6_finseq_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,spc1_numerals,spc1_boole,t2_tarski,fraenkel_a_2_1_graph_2,s5_nat_1__e2_7__graph_2,e1_7__graph_2]), [interesting(0.8),file(graph_2,e2_7__graph_2),[file(graph_2,e2_7__graph_2)]]). fof(dt_c4_7__graph_2,plain,( m2_subset_1(c4_7__graph_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dh_c4_7__graph_2,e2_7__graph_2]), [interesting(0.8),file(graph_2,c4_7__graph_2),[file(graph_2,c4_7__graph_2)]]). 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(rqLessOrEqual__r1_xreal_0__r1_r1,theorem,( r1_xreal_0(1,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r1)]). fof(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_c6_7__graph_2,definition,( c6_7__graph_2 = a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2) ), introduced(definition,[new_symbol(c6_7__graph_2),file(graph_2,c6_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c6_7__graph_2)]). fof(dt_c1_7_1__graph_2,assumption,( $true ), introduced(assumption,[file(graph_2,c1_7_1__graph_2)]), [interesting(0.65),axiom,file(graph_2,c1_7_1__graph_2)]). fof(d3_tarski,definition,( ! [A,B] : ( r1_tarski(A,B) <=> ! [C] : ( r2_hidden(C,A) => r2_hidden(C,B) ) ) ), file(tarski,d3_tarski), [interesting(0.9),axiom,file(tarski,d3_tarski)]). fof(dh_c1_7_1__graph_2,definition, ( ~ ( r2_hidden(c1_7_1__graph_2,a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)) & ~ r2_hidden(c1_7_1__graph_2,k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ) => ! [A] : ~ ( r2_hidden(A,a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)) & ~ r2_hidden(A,k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ) ), introduced(definition,[new_symbol(c1_7_1__graph_2),file(graph_2,c1_7_1__graph_2)]), [interesting(0.65),axiom,file(graph_2,c1_7_1__graph_2)]). fof(e1_7_1__graph_2,assumption,( r2_hidden(c1_7_1__graph_2,a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)) ), introduced(assumption,[file(graph_2,e1_7_1__graph_2)]), [interesting(0.65),axiom,file(graph_2,e1_7_1__graph_2)]). fof(dh_c2_7_1__graph_2,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & A = c1_7_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),A) & r1_xreal_0(A,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) => ( m2_subset_1(c2_7_1__graph_2,k1_numbers,k5_numbers) & c2_7_1__graph_2 = c1_7_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),c2_7_1__graph_2) & r1_xreal_0(c2_7_1__graph_2,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) ), introduced(definition,[new_symbol(c2_7_1__graph_2),file(graph_2,c2_7_1__graph_2)]), [interesting(0.65),axiom,file(graph_2,c2_7_1__graph_2)]). fof(e2_7_1__graph_2,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & A = c1_7_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),A) & r1_xreal_0(A,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,e1_7_1__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e1_7_1__graph_2]), [interesting(0.65),file(graph_2,e2_7_1__graph_2),[file(graph_2,e2_7_1__graph_2)]]). fof(dt_c2_7_1__graph_2,plain,( m2_subset_1(c2_7_1__graph_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,e1_7_1__graph_2])],[dh_c2_7_1__graph_2,e2_7_1__graph_2]), [interesting(0.65),file(graph_2,c2_7_1__graph_2),[file(graph_2,c2_7_1__graph_2)]]). fof(t29_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ), file(nat_1,t29_nat_1), [interesting(0.9),axiom,file(nat_1,t29_nat_1)]). fof(e4_7_1__graph_2,plain,( r1_xreal_0(1,k1_nat_1(c2_7__graph_2,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c2_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_numerals,spc1_boole,t29_nat_1]), [interesting(0.65),file(graph_2,e4_7_1__graph_2),[file(graph_2,e4_7_1__graph_2)]]). fof(e3_7_1__graph_2,plain, ( c2_7_1__graph_2 = c1_7_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),c2_7_1__graph_2) & r1_xreal_0(c2_7_1__graph_2,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,e1_7_1__graph_2])],[dh_c2_7_1__graph_2,e2_7_1__graph_2]), [interesting(0.65),file(graph_2,e3_7_1__graph_2),[file(graph_2,e3_7_1__graph_2)]]). fof(t2_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( ( r1_xreal_0(A,B) & r1_xreal_0(B,C) ) => r1_xreal_0(A,C) ) ) ) ) ), file(xreal_1,t2_xreal_1), [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]). fof(e5_7_1__graph_2,plain,( r1_xreal_0(1,c2_7_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,e1_7_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,dt_c2_7_1__graph_2,cc2_xreal_0,spc1_numerals,spc1_boole,e4_7_1__graph_2,e3_7_1__graph_2,t2_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(graph_2,e5_7_1__graph_2),[file(graph_2,e5_7_1__graph_2)]]). fof(t3_finseq_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(A,k2_finseq_1(B)) <=> ( r1_xreal_0(1,A) & r1_xreal_0(A,B) ) ) ) ) ), file(finseq_1,t3_finseq_1), [interesting(0.9),axiom,file(finseq_1,t3_finseq_1)]). fof(e6_7_1__graph_2,plain,( r2_hidden(c1_7_1__graph_2,k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,e1_7_1__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k2_finseq_1,dt_k1_nat_1,dt_k2_finseq_1,dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,dt_c2_7_1__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,t1_subset,t7_boole,spc1_numerals,spc1_boole,e5_7_1__graph_2,e3_7_1__graph_2,t3_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(graph_2,e6_7_1__graph_2),[file(graph_2,e6_7_1__graph_2)]]). fof(i3_7_1__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i3_7_1__graph_2)]), [interesting(0.65),trivial,file(graph_2,i3_7_1__graph_2)]). fof(i2_7_1__graph_2,plain,( r2_hidden(c1_7_1__graph_2,k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ), inference(conclusion,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2,e1_7_1__graph_2])],[e6_7_1__graph_2,i3_7_1__graph_2]), [interesting(0.65),file(graph_2,i2_7_1__graph_2),[file(graph_2,i2_7_1__graph_2)]]). fof(i1_7_1__graph_2,plain,( ~ ( r2_hidden(c1_7_1__graph_2,a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)) & ~ r2_hidden(c1_7_1__graph_2,k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_1__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[e1_7_1__graph_2])],[e1_7_1__graph_2,i2_7_1__graph_2]), [interesting(0.65),file(graph_2,i1_7_1__graph_2),[file(graph_2,i1_7_1__graph_2)]]). fof(i1_7_1_tmp__graph_2,plain,( ~ ( r2_hidden(c1_7_1__graph_2,a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)) & ~ r2_hidden(c1_7_1__graph_2,k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[dt_c1_7_1__graph_2])],[dt_c1_7_1__graph_2,i1_7_1__graph_2]), [interesting(0.8),e5_7__graph_2]). fof(e5_7__graph_2,plain,( r1_tarski(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2),k2_finseq_1(k1_nat_1(c2_7__graph_2,c1_7__graph_2))) ), inference(let,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[i1_7_1_tmp__graph_2,dt_k5_ordinal2,cc1_card_1,cc1_finset_1,cc1_funct_1,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_ordinal2,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_nat_1,cc1_xreal_0,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_int_1,cc3_nat_1,fc1_finseq_1,fc1_nat_1,spc1_numerals,spc1_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k2_finseq_1,dt_k1_nat_1,dt_k2_finseq_1,dt_c1_7__graph_2,dt_c2_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,d3_tarski,dh_c1_7_1__graph_2]), [interesting(0.8),file(graph_2,e5_7__graph_2),[file(graph_2,e5_7__graph_2)]]). fof(t13_finset_1,theorem,( ! [A,B] : ( ( r1_tarski(A,B) & v1_finset_1(B) ) => v1_finset_1(A) ) ), file(finset_1,t13_finset_1), [interesting(0.9),axiom,file(finset_1,t13_finset_1)]). fof(e6_7__graph_2,plain,( v1_finset_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,fc2_finseq_1,dt_k5_ordinal2,cc1_card_1,cc1_finset_1,cc1_funct_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_ordinal2,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k2_xcmplx_0,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_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_nat_1,cc1_xreal_0,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_int_1,cc3_nat_1,fc1_finseq_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t4_subset,t7_boole,spc1_numerals,spc1_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,redefinition_k1_nat_1,redefinition_k2_finseq_1,dt_k1_nat_1,dt_k2_finseq_1,dt_c1_7__graph_2,dt_c2_7__graph_2,t3_subset,t2_tarski,fraenkel_a_2_1_graph_2,e5_7__graph_2,t13_finset_1]), [interesting(0.8),file(graph_2,e6_7__graph_2),[file(graph_2,e6_7__graph_2)]]). fof(dt_c6_7__graph_2,plain,( v1_finset_1(c6_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_card_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,rc1_finset_1,rc3_finset_1,rc4_finset_1,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,spc1_numerals,spc1_boole,dt_c1_7__graph_2,dt_c2_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,de_c6_7__graph_2,e6_7__graph_2]), [interesting(0.8),file(graph_2,c6_7__graph_2),[file(graph_2,c6_7__graph_2)]]). fof(dh_c7_7__graph_2,definition, ( ? [A] : ( r2_hidden(A,k4_finseq_1(k14_finseq_1(c6_7__graph_2))) & k1_nat_1(c2_7__graph_2,c4_7__graph_2) = k1_funct_1(k14_finseq_1(c6_7__graph_2),A) ) => ( r2_hidden(c7_7__graph_2,k4_finseq_1(k14_finseq_1(c6_7__graph_2))) & k1_nat_1(c2_7__graph_2,c4_7__graph_2) = k1_funct_1(k14_finseq_1(c6_7__graph_2),c7_7__graph_2) ) ), introduced(definition,[new_symbol(c7_7__graph_2),file(graph_2,c7_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c7_7__graph_2)]). fof(e3_7__graph_2,plain, ( r1_xreal_0(1,c4_7__graph_2) & r1_xreal_0(c4_7__graph_2,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c4_7__graph_2) != k1_nat_1(c2_7__graph_2,c4_7__graph_2) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dh_c4_7__graph_2,e2_7__graph_2]), [interesting(0.8),file(graph_2,e3_7__graph_2),[file(graph_2,e3_7__graph_2)]]). fof(t9_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ! [D] : ( v1_xreal_0(D) => ( ( r1_xreal_0(A,B) & r1_xreal_0(C,D) ) => r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,D)) ) ) ) ) ) ), file(xreal_1,t9_xreal_1), [interesting(0.9),axiom,file(xreal_1,t9_xreal_1)]). fof(e8_7__graph_2,plain, ( r1_xreal_0(k1_nat_1(c2_7__graph_2,1),k1_nat_1(c2_7__graph_2,c4_7__graph_2)) & r1_xreal_0(k1_nat_1(c2_7__graph_2,c4_7__graph_2),k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc8_xreal_0,rc1_xreal_0,spc6_arithm,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_recdef_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,cc2_xreal_0,fc3_xreal_0,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e3_7__graph_2,t9_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(graph_2,e8_7__graph_2),[file(graph_2,e8_7__graph_2)]]). fof(e9_7__graph_2,plain,( r2_hidden(k1_nat_1(c2_7__graph_2,c4_7__graph_2),c6_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_finset_1,t1_real,t2_subset,t4_real,t6_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,dt_k1_nat_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,t1_subset,t7_boole,spc1_numerals,spc1_boole,e8_7__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(graph_2,e9_7__graph_2),[file(graph_2,e9_7__graph_2)]]). fof(d13_finseq_1,definition,( ! [A] : ( ? [B] : ( v4_ordinal2(B) & r1_tarski(A,k2_finseq_1(B)) ) => ! [B] : ( m2_finseq_1(B,k5_numbers) => ( B = k14_finseq_1(A) <=> ( k2_relat_1(B) = A & ! [C] : ( v4_ordinal2(C) => ! [D] : ( v4_ordinal2(D) => ! [E] : ( v4_ordinal2(E) => ! [F] : ( v4_ordinal2(F) => ~ ( r1_xreal_0(1,C) & ~ r1_xreal_0(D,C) & r1_xreal_0(D,k3_finseq_1(B)) & E = k1_funct_1(B,C) & F = k1_funct_1(B,D) & r1_xreal_0(F,E) ) ) ) ) ) ) ) ) ) ), file(finseq_1,d13_finseq_1), [interesting(0.9),axiom,file(finseq_1,d13_finseq_1)]). fof(e13_7__graph_2,plain,( r2_hidden(k1_nat_1(c2_7__graph_2,c4_7__graph_2),k2_relat_1(k14_finseq_1(c6_7__graph_2))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[rc4_funct_1,existence_m1_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_card_1,cc1_relset_1,fc14_finset_1,fc2_finseq_1,rc1_card_1,rc2_card_1,rc2_finseq_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,commutativity_k2_xcmplx_0,existence_m1_finseq_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_card_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k5_numbers,redefinition_m2_finseq_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k2_finseq_1,dt_k2_relat_1,dt_k3_finseq_1,dt_k5_numbers,dt_m2_finseq_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t3_subset,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e5_7__graph_2,e9_7__graph_2,d13_finseq_1]), [interesting(0.8),file(graph_2,e13_7__graph_2),[file(graph_2,e13_7__graph_2)]]). fof(d5_funct_1,definition,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) ) => ! [B] : ( B = k2_relat_1(A) <=> ! [C] : ( r2_hidden(C,B) <=> ? [D] : ( r2_hidden(D,k1_relat_1(A)) & C = k1_funct_1(A,D) ) ) ) ) ), file(funct_1,d5_funct_1), [interesting(0.9),axiom,file(funct_1,d5_funct_1)]). fof(e14_7__graph_2,plain,( ? [A] : ( r2_hidden(A,k4_finseq_1(k14_finseq_1(c6_7__graph_2))) & k1_nat_1(c2_7__graph_2,c4_7__graph_2) = k1_funct_1(k14_finseq_1(c6_7__graph_2),A) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,cc3_xreal_0,cc6_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_finset_1,fc8_xreal_0,fc9_xreal_0,rc2_finseq_1,rc2_xreal_0,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,t1_real,t4_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc6_arithm,t2_real,t3_real,spc1_numerals,spc1_boole,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,dt_c1_7__graph_2,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc11_finseq_1,fc17_finseq_1,rc1_finseq_1,rc1_finset_1,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k4_finseq_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_relat_1,dt_k2_relat_1,dt_k4_finseq_1,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,rc1_funct_1,t1_subset,t7_boole,e13_7__graph_2,d5_funct_1]), [interesting(0.8),file(graph_2,e14_7__graph_2),[file(graph_2,e14_7__graph_2)]]). fof(dt_c7_7__graph_2,plain,( $true ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dh_c7_7__graph_2,e14_7__graph_2]), [interesting(0.8),file(graph_2,c7_7__graph_2),[file(graph_2,c7_7__graph_2)]]). fof(de_c8_7__graph_2,definition,( c8_7__graph_2 = c7_7__graph_2 ), introduced(definition,[new_symbol(c8_7__graph_2),file(graph_2,c8_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c8_7__graph_2)]). 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(rqLessOrEqual__r1_xreal_0__r0_r0,theorem,( r1_xreal_0(0,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r0)]). fof(rqLessOrEqual__r1_xreal_0__r0_r1,theorem,( r1_xreal_0(0,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r1)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm1,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r1_r0,theorem,( ~ r1_xreal_0(1,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r0)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm1,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r0,theorem,( r1_xreal_0(k4_xcmplx_0(1),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r1,theorem,( r1_xreal_0(k4_xcmplx_0(1),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm1)]). fof(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(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(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(e15_7__graph_2,plain, ( r2_hidden(c7_7__graph_2,k4_finseq_1(k14_finseq_1(c6_7__graph_2))) & k1_nat_1(c2_7__graph_2,c4_7__graph_2) = k1_funct_1(k14_finseq_1(c6_7__graph_2),c7_7__graph_2) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dh_c7_7__graph_2,e14_7__graph_2]), [interesting(0.8),file(graph_2,e15_7__graph_2),[file(graph_2,e15_7__graph_2)]]). fof(rqRealNeg__k4_xcmplx_0__r1_rm1,theorem,( k4_xcmplx_0(1) = k4_xcmplx_0(1) ), file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1), [interesting(0.9),axiom,file(arithm,rqRealNeg__k4_xcmplx_0__r1_rm1)]). fof(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(e16_7__graph_2,plain,( m2_subset_1(c7_7__graph_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_finset_1,rc2_finseq_1,rc4_funct_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_xreal_0,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_finseq_1,fc30_xreal_0,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_xreal_0,fc6_int_1,fc6_xreal_0,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_finseq_1,rc4_xreal_0,rc6_finseq_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,dt_m2_finseq_1,dt_c1_7__graph_2,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc17_finseq_1,fc1_ordinal2,fc5_int_1,fc9_int_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc2_funct_1,rc3_finset_1,rc4_finset_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,t3_subset,t4_arithm,t4_subset,t5_arithm,t5_subset,t6_arithm,t6_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k4_finseq_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_numbers,dt_k4_finseq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_m2_subset_1,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,dt_c7_7__graph_2,de_c6_7__graph_2,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,t7_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e15_7__graph_2,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(0.8),file(graph_2,e16_7__graph_2),[file(graph_2,e16_7__graph_2)]]). fof(dt_c8_7__graph_2,plain,( m2_subset_1(c8_7__graph_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[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_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_xreal_0,rc4_funct_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc7_xreal_0,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc3_finset_1,rc3_nat_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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c7_7__graph_2,de_c8_7__graph_2,e16_7__graph_2]), [interesting(0.8),file(graph_2,c8_7__graph_2),[file(graph_2,c8_7__graph_2)]]). fof(de_c9_7__graph_2,definition,( c9_7__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c4_7__graph_2) ), introduced(definition,[new_symbol(c9_7__graph_2),file(graph_2,c9_7__graph_2)]), [interesting(0.8),axiom,file(graph_2,c9_7__graph_2)]). fof(e17_7__graph_2,plain,( m2_subset_1(k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c4_7__graph_2),k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_numerals,spc1_boole,existence_m2_subset_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2]), [interesting(0.8),file(graph_2,e17_7__graph_2),[file(graph_2,e17_7__graph_2)]]). fof(dt_c9_7__graph_2,plain,( m2_subset_1(c9_7__graph_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_numerals,spc1_boole,existence_m2_subset_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,de_c9_7__graph_2,e17_7__graph_2]), [interesting(0.8),file(graph_2,c9_7__graph_2),[file(graph_2,c9_7__graph_2)]]). fof(rqLessOrEqual__r1_xreal_0__r0_r2,theorem,( r1_xreal_0(0,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_r2)]). fof(rqLessOrEqual__r1_xreal_0__r0_rm2,theorem,( ~ r1_xreal_0(0,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r0_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r1_r2,theorem,( r1_xreal_0(1,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_r2)]). fof(rqLessOrEqual__r1_xreal_0__r1_rm2,theorem,( ~ r1_xreal_0(1,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__r2_r0,theorem,( ~ r1_xreal_0(2,0) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r0)]). fof(rqLessOrEqual__r1_xreal_0__r2_r1,theorem,( ~ r1_xreal_0(2,1) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r1)]). fof(rqLessOrEqual__r1_xreal_0__r2_r2,theorem,( r1_xreal_0(2,2) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_r2)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm1,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__r2_rm2,theorem,( ~ r1_xreal_0(2,k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__r2_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_r2,theorem,( r1_xreal_0(k4_xcmplx_0(1),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm1_rm2,theorem,( ~ r1_xreal_0(k4_xcmplx_0(1),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm1_rm2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r0,theorem,( r1_xreal_0(k4_xcmplx_0(2),0) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r0)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r1,theorem,( r1_xreal_0(k4_xcmplx_0(2),1) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_r2,theorem,( r1_xreal_0(k4_xcmplx_0(2),2) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_r2)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm1,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(1)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm1)]). fof(rqLessOrEqual__r1_xreal_0__rm2_rm2,theorem,( r1_xreal_0(k4_xcmplx_0(2),k4_xcmplx_0(2)) ), file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2), [interesting(0.9),axiom,file(arithm,rqLessOrEqual__r1_xreal_0__rm2_rm2)]). 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(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(redefinition_k4_card_1,definition,( ! [A] : ( v1_finset_1(A) => k4_card_1(A) = k1_card_1(A) ) ), file(card_1,k4_card_1), [interesting(0.9),axiom,file(card_1,k4_card_1)]). fof(dt_k4_card_1,axiom,( ! [A] : ( v1_finset_1(A) => m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) ), file(card_1,k4_card_1), [interesting(0.9),axiom,file(card_1,k4_card_1)]). fof(e1_7_2_1_1__graph_2,assumption,( c1_7__graph_2 = 0 ), introduced(assumption,[file(graph_2,e1_7_2_1_1__graph_2)]), [interesting(0.35),axiom,file(graph_2,e1_7_2_1_1__graph_2)]). fof(e1_7_2_1_1_1__graph_2,assumption,( c6_7__graph_2 != k1_xboole_0 ), introduced(assumption,[file(graph_2,e1_7_2_1_1_1__graph_2)]), [interesting(0.2),axiom,file(graph_2,e1_7_2_1_1_1__graph_2)]). fof(dh_c1_7_2_1_1_1__graph_2,definition, ( ? [A] : r2_hidden(A,c6_7__graph_2) => r2_hidden(c1_7_2_1_1_1__graph_2,c6_7__graph_2) ), introduced(definition,[new_symbol(c1_7_2_1_1_1__graph_2),file(graph_2,c1_7_2_1_1_1__graph_2)]), [interesting(0.2),axiom,file(graph_2,c1_7_2_1_1_1__graph_2)]). fof(d1_xboole_0,definition,( ! [A] : ( A = k1_xboole_0 <=> ! [B] : ~ r2_hidden(B,A) ) ), file(xboole_0,d1_xboole_0), [interesting(0.9),axiom,file(xboole_0,d1_xboole_0)]). fof(e2_7_2_1_1_1__graph_2,plain,( ? [A] : r2_hidden(A,c6_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[reflexivity_r1_tarski,commutativity_k2_xcmplx_0,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc1_numerals,spc1_boole,existence_m1_subset_1,dt_m1_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc2_funct_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_c6_7__graph_2,de_c6_7__graph_2,fc2_finseq_1,t1_subset,t6_boole,t7_boole,e1_7_2_1_1_1__graph_2,d1_xboole_0]), [interesting(0.2),file(graph_2,e2_7_2_1_1_1__graph_2),[file(graph_2,e2_7_2_1_1_1__graph_2)]]). fof(dt_c1_7_2_1_1_1__graph_2,plain,( $true ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[dh_c1_7_2_1_1_1__graph_2,e2_7_2_1_1_1__graph_2]), [interesting(0.2),file(graph_2,c1_7_2_1_1_1__graph_2),[file(graph_2,c1_7_2_1_1_1__graph_2)]]). fof(dh_c2_7_2_1_1_1__graph_2,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & A = c1_7_2_1_1_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),A) & r1_xreal_0(A,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) => ( m2_subset_1(c2_7_2_1_1_1__graph_2,k1_numbers,k5_numbers) & c2_7_2_1_1_1__graph_2 = c1_7_2_1_1_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),c2_7_2_1_1_1__graph_2) & r1_xreal_0(c2_7_2_1_1_1__graph_2,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) ), introduced(definition,[new_symbol(c2_7_2_1_1_1__graph_2),file(graph_2,c2_7_2_1_1_1__graph_2)]), [interesting(0.2),axiom,file(graph_2,c2_7_2_1_1_1__graph_2)]). fof(e3_7_2_1_1_1__graph_2,plain,( r2_hidden(c1_7_2_1_1_1__graph_2,c6_7__graph_2) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[dh_c1_7_2_1_1_1__graph_2,e2_7_2_1_1_1__graph_2]), [interesting(0.2),file(graph_2,e3_7_2_1_1_1__graph_2),[file(graph_2,e3_7_2_1_1_1__graph_2)]]). fof(e4_7_2_1_1_1__graph_2,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & A = c1_7_2_1_1_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),A) & r1_xreal_0(A,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_card_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,rc1_finset_1,rc3_finset_1,rc4_finset_1,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c1_7_2_1_1_1__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1,t1_subset,t7_boole,spc1_numerals,spc1_boole,e3_7_2_1_1_1__graph_2]), [interesting(0.2),file(graph_2,e4_7_2_1_1_1__graph_2),[file(graph_2,e4_7_2_1_1_1__graph_2)]]). fof(dt_c2_7_2_1_1_1__graph_2,plain,( m2_subset_1(c2_7_2_1_1_1__graph_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[dh_c2_7_2_1_1_1__graph_2,e4_7_2_1_1_1__graph_2]), [interesting(0.2),file(graph_2,c2_7_2_1_1_1__graph_2),[file(graph_2,c2_7_2_1_1_1__graph_2)]]). fof(e5_7_2_1_1_1__graph_2,plain, ( c2_7_2_1_1_1__graph_2 = c1_7_2_1_1_1__graph_2 & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),c2_7_2_1_1_1__graph_2) & r1_xreal_0(c2_7_2_1_1_1__graph_2,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[dh_c2_7_2_1_1_1__graph_2,e4_7_2_1_1_1__graph_2]), [interesting(0.2),file(graph_2,e5_7_2_1_1_1__graph_2),[file(graph_2,e5_7_2_1_1_1__graph_2)]]). fof(e6_7_2_1_1_1__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,1),k1_nat_1(c2_7__graph_2,0)) ), inference(mizar_by,[status(thm),assumptions([e1_7_2_1_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_c1_7__graph_2,dt_c1_7_2_1_1_1__graph_2,dt_c2_7__graph_2,dt_c2_7_2_1_1_1__graph_2,cc2_xreal_0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_7_2_1_1__graph_2,e5_7_2_1_1_1__graph_2,t2_xreal_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.2),file(graph_2,e6_7_2_1_1_1__graph_2),[file(graph_2,e6_7_2_1_1_1__graph_2)]]). fof(t8_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k2_xcmplx_0(A,C),k2_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t8_xreal_1), [interesting(0.9),axiom,file(xreal_1,t8_xreal_1)]). fof(e7_7_2_1_1_1__graph_2,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([e1_7_2_1_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc8_xreal_0,rc1_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c2_7__graph_2,cc2_xreal_0,fc3_xreal_0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e6_7_2_1_1_1__graph_2,t8_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.2),file(graph_2,e7_7_2_1_1_1__graph_2),[file(graph_2,e7_7_2_1_1_1__graph_2)]]). fof(i2_7_2_1_1_1__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_2_1_1_1__graph_2)]), [interesting(0.2),trivial,file(graph_2,i2_7_2_1_1_1__graph_2)]). fof(i1_7_2_1_1_1__graph_2,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([e1_7_2_1_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1_1__graph_2])],[e7_7_2_1_1_1__graph_2,i2_7_2_1_1_1__graph_2]), [interesting(0.2),file(graph_2,i1_7_2_1_1_1__graph_2),[file(graph_2,i1_7_2_1_1_1__graph_2)]]). fof(e2_7_2_1_1__graph_2,plain,( c6_7__graph_2 = k1_xboole_0 ), inference(discharge_asm,[status(thm),assumptions([e1_7_2_1_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[e1_7_2_1_1_1__graph_2])],[e1_7_2_1_1_1__graph_2,i1_7_2_1_1_1__graph_2]), [interesting(0.35),file(graph_2,e2_7_2_1_1__graph_2),[file(graph_2,e2_7_2_1_1__graph_2)]]). fof(t78_card_1,theorem,( k4_card_1(k1_xboole_0) = 0 ), file(card_1,t78_card_1), [interesting(0.9),axiom,file(card_1,t78_card_1)]). fof(e3_7_2_1_1__graph_2,plain,( k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1__graph_2])],[commutativity_k2_xcmplx_0,reflexivity_r1_tarski,dt_k2_xcmplx_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,spc6_arithm,t1_arithm,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc1_ordinal2,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,spc1_numerals,t1_real,t1_subset,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,spc1_numerals,spc1_boole,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c2_7__graph_2,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc2_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t1_numerals,t2_subset,t7_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,redefinition_k4_card_1,dt_k1_xboole_0,dt_k4_card_1,dt_c1_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,fc2_finseq_1,t6_boole,spc0_numerals,spc0_boole,e2_7_2_1_1__graph_2,e1_7_2_1_1__graph_2,t78_card_1]), [interesting(0.35),file(graph_2,e3_7_2_1_1__graph_2),[file(graph_2,e3_7_2_1_1__graph_2)]]). fof(i2_7_2_1_1__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_2_1_1__graph_2)]), [interesting(0.35),trivial,file(graph_2,i2_7_2_1_1__graph_2)]). fof(i1_7_2_1_1__graph_2,plain,( k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(conclusion,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7_2_1_1__graph_2])],[e3_7_2_1_1__graph_2,i2_7_2_1_1__graph_2]), [interesting(0.35),file(graph_2,i1_7_2_1_1__graph_2),[file(graph_2,i1_7_2_1_1__graph_2)]]). fof(i1_7_2_1__graph_2,plain, ( c1_7__graph_2 = 0 => k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[e1_7_2_1_1__graph_2])],[e1_7_2_1_1__graph_2,i1_7_2_1_1__graph_2]), [interesting(0.5),file(graph_2,i1_7_2_1__graph_2),[file(graph_2,i1_7_2_1__graph_2)]]). fof(e1_7_2_1_2__graph_2,assumption,( ~ r1_xreal_0(c1_7__graph_2,0) ), introduced(assumption,[file(graph_2,e1_7_2_1_2__graph_2)]), [interesting(0.35),axiom,file(graph_2,e1_7_2_1_2__graph_2)]). fof(de_c1_7_2_1_2__graph_2,definition,( c1_7_2_1_2__graph_2 = k6_xcmplx_0(c1_7__graph_2,1) ), introduced(definition,[new_symbol(c1_7_2_1_2__graph_2),file(graph_2,c1_7_2_1_2__graph_2)]), [interesting(0.35),axiom,file(graph_2,c1_7_2_1_2__graph_2)]). fof(dt_k5_binarith,axiom,( ! [A,B] : ( ( v4_ordinal2(A) & v4_ordinal2(B) ) => m2_subset_1(k5_binarith(A,B),k1_numbers,k5_numbers) ) ), file(binarith,k5_binarith), [interesting(0.9),axiom,file(binarith,k5_binarith)]). fof(t38_nat_1,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A) <=> r1_xreal_0(A,B) ) ) ) ), file(nat_1,t38_nat_1), [interesting(0.9),axiom,file(nat_1,t38_nat_1)]). fof(e2_7_2_1_2__graph_2,plain,( r1_xreal_0(k1_nat_1(0,1),c1_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_arithm,t1_numerals,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c1_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_7_2_1_2__graph_2,t38_nat_1,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0]), [interesting(0.35),file(graph_2,e2_7_2_1_2__graph_2),[file(graph_2,e2_7_2_1_2__graph_2)]]). fof(t11_xreal_1,theorem,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ! [C] : ( v1_xreal_0(C) => ( r1_xreal_0(A,B) <=> r1_xreal_0(k6_xcmplx_0(A,C),k6_xcmplx_0(B,C)) ) ) ) ) ), file(xreal_1,t11_xreal_1), [interesting(0.9),axiom,file(xreal_1,t11_xreal_1)]). fof(e3_7_2_1_2__graph_2,plain,( r1_xreal_0(k6_xcmplx_0(1,1),k6_xcmplx_0(c1_7__graph_2,1)) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc5_int_1,fc8_xreal_0,fc9_int_1,rc1_xreal_0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_subset,t4_arithm,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__graph_2,cc2_xreal_0,fc1_xreal_0,fc3_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_rm1_rm1,rqRealAdd__k2_xcmplx_0__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e2_7_2_1_2__graph_2,t11_xreal_1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r0]), [interesting(0.35),file(graph_2,e3_7_2_1_2__graph_2),[file(graph_2,e3_7_2_1_2__graph_2)]]). fof(d3_binarith,definition,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v4_ordinal2(B) => ( ( r1_xreal_0(0,k6_xcmplx_0(A,B)) => k5_binarith(A,B) = k6_xcmplx_0(A,B) ) & ( ~ r1_xreal_0(0,k6_xcmplx_0(A,B)) => k5_binarith(A,B) = 0 ) ) ) ) ), file(binarith,d3_binarith), [interesting(0.9),axiom,file(binarith,d3_binarith)]). fof(e4_7_2_1_2__graph_2,plain,( k5_binarith(c1_7__graph_2,1) = k6_xcmplx_0(c1_7__graph_2,1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_xreal_0,fc20_xreal_0,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_int_1,fc5_xreal_0,fc6_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_real,t2_real,t2_subset,t3_real,t4_arithm,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k5_binarith,dt_k6_xcmplx_0,dt_c1_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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__rm1_r0_rm1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,rqRealDiff__k6_xcmplx_0__r0_rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealNeg__k4_xcmplx_0__r0_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e3_7_2_1_2__graph_2,d3_binarith,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealAdd__k2_xcmplx_0__r1_rm1_r0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1]), [interesting(0.35),file(graph_2,e4_7_2_1_2__graph_2),[file(graph_2,e4_7_2_1_2__graph_2)]]). 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(e5_7_2_1_2__graph_2,plain,( m2_subset_1(k6_xcmplx_0(c1_7__graph_2,1),k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_card_1,cc2_finset_1,cc2_xreal_0,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,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,fc2_finseq_1,fc3_int_1,fc4_int_1,fc5_xreal_0,fc8_int_1,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc1_xreal_0,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_int_1,cc3_nat_1,fc1_ordinal2,fc5_int_1,fc9_int_1,spc9_arithm,t1_numerals,t2_subset,t3_subset,t4_arithm,t6_boole,t7_boole,t8_boole,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k4_xcmplx_0,dt_k5_binarith,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_7__graph_2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e4_7_2_1_2__graph_2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2]), [interesting(0.35),file(graph_2,e5_7_2_1_2__graph_2),[file(graph_2,e5_7_2_1_2__graph_2)]]). fof(dt_c1_7_2_1_2__graph_2,plain,( m2_subset_1(c1_7_2_1_2__graph_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k1_xboole_0,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc20_xreal_0,fc2_finseq_1,fc4_int_1,fc5_xreal_0,fc8_int_1,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t4_subset,t5_subset,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,fc9_int_1,t2_subset,t3_subset,t6_boole,t7_boole,t8_boole,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_7__graph_2,spc1_numerals,spc1_boole,de_c1_7_2_1_2__graph_2,e5_7_2_1_2__graph_2]), [interesting(0.35),file(graph_2,c1_7_2_1_2__graph_2),[file(graph_2,c1_7_2_1_2__graph_2)]]). fof(fraenkel_a_2_0_graph_2,definition,( ! [A,B,C] : ( ( m2_subset_1(B,k1_numbers,k5_numbers) & m2_subset_1(C,k1_numbers,k5_numbers) ) => ( r2_hidden(A,a_2_0_graph_2(B,C)) <=> ? [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) & A = D & r1_xreal_0(B,D) & r1_xreal_0(D,k1_nat_1(B,C)) ) ) ) ), file(graph_2,a_2_0_graph_2), [interesting(0.9),axiom,file(graph_2,a_2_0_graph_2)]). fof(e6_7_2_1_2__graph_2,plain,( k1_nat_1(k1_nat_1(c2_7__graph_2,1),c1_7_2_1_2__graph_2) = k1_nat_1(c2_7__graph_2,c1_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2,dt_c2_7__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc1_xreal_0,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_xreal_0,fc6_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc5_int_1,fc9_int_1,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t4_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__graph_2,dt_c1_7_2_1_2__graph_2,dt_c2_7__graph_2,de_c1_7_2_1_2__graph_2,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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealAdd__k2_xcmplx_0__r1_rm1_r0]), [interesting(0.35),file(graph_2,e6_7_2_1_2__graph_2),[file(graph_2,e6_7_2_1_2__graph_2)]]). fof(t4_graph_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => k1_card_1(a_2_0_graph_2(A,B)) = k1_nat_1(B,1) ) ) ), file(graph_2,t4_graph_2), [interesting(0.9),axiom,file(graph_2,t4_graph_2)]). fof(e7_7_2_1_2__graph_2,plain,( k4_card_1(c6_7__graph_2) = k1_nat_1(c1_7_2_1_2__graph_2,1) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2,dt_c2_7__graph_2])],[cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,cc1_xreal_0,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_xreal_0,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_xreal_0,fc6_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_nat_1,rc3_xreal_0,rc4_xreal_0,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_subset_1,cc1_card_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,fc2_card_1,fc5_int_1,fc9_int_1,rc1_card_1,rc1_finset_1,rc3_finset_1,rc4_finset_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t1_subset,t2_subset,t3_subset,t4_arithm,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k4_card_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_nat_1,dt_k1_numbers,dt_k2_xcmplx_0,dt_k4_card_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_subset_1,dt_c1_7__graph_2,dt_c1_7_2_1_2__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c1_7_2_1_2__graph_2,de_c6_7__graph_2,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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r2_r0_r2,rqRealDiff__k6_xcmplx_0__r2_r2_r0,rqRealDiff__k6_xcmplx_0__rm1_r0_rm1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,t2_tarski,fraenkel_a_2_0_graph_2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e6_7_2_1_2__graph_2,t4_graph_2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealAdd__k2_xcmplx_0__rm1_r1_r0]), [interesting(0.35),file(graph_2,e7_7_2_1_2__graph_2),[file(graph_2,e7_7_2_1_2__graph_2)]]). fof(e8_7_2_1_2__graph_2,plain,( k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2,dt_c2_7__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc1_xreal_0,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_xreal_0,fc6_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,t1_real,t1_subset,t2_real,t3_real,t3_subset,t4_real,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c2_7__graph_2,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc2_card_1,fc5_int_1,fc9_int_1,rc1_finset_1,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_numerals,t2_subset,t4_arithm,t6_boole,t7_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,redefinition_k1_nat_1,redefinition_k4_card_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_k4_card_1,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_c1_7__graph_2,dt_c1_7_2_1_2__graph_2,dt_c6_7__graph_2,de_c1_7_2_1_2__graph_2,de_c6_7__graph_2,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_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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,rqRealDiff__k6_xcmplx_0__r2_r0_r2,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e7_7_2_1_2__graph_2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__r1_rm1_r2,rqRealDiff__k6_xcmplx_0__r2_r1_r1,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealAdd__k2_xcmplx_0__rm1_r1_r0]), [interesting(0.35),file(graph_2,e8_7_2_1_2__graph_2),[file(graph_2,e8_7_2_1_2__graph_2)]]). fof(i2_7_2_1_2__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_2_1_2__graph_2)]), [interesting(0.35),trivial,file(graph_2,i2_7_2_1_2__graph_2)]). fof(i1_7_2_1_2__graph_2,plain,( k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(conclusion,[status(thm),assumptions([dt_c1_7__graph_2,e1_7_2_1_2__graph_2,dt_c2_7__graph_2])],[e8_7_2_1_2__graph_2,i2_7_2_1_2__graph_2]), [interesting(0.35),file(graph_2,i1_7_2_1_2__graph_2),[file(graph_2,i1_7_2_1_2__graph_2)]]). fof(i2_7_2_1__graph_2,plain, ( ~ r1_xreal_0(c1_7__graph_2,0) => k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[e1_7_2_1_2__graph_2])],[e1_7_2_1_2__graph_2,i1_7_2_1_2__graph_2]), [interesting(0.5),file(graph_2,i2_7_2_1__graph_2),[file(graph_2,i2_7_2_1__graph_2)]]). fof(e1_7_2_1__graph_2,plain,( ~ ( c1_7__graph_2 != 0 & r1_xreal_0(c1_7__graph_2,0) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,t1_numerals,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_c1_7__graph_2,rqLessOrEqual__r1_xreal_0__r0_r0,spc0_numerals,spc0_boole]), [interesting(0.5),file(graph_2,e1_7_2_1__graph_2),[file(graph_2,e1_7_2_1__graph_2)]]). fof(e7_7__graph_2,plain,( k4_card_1(c6_7__graph_2) = c1_7__graph_2 ), inference(percases,[status(thm),assumptions([dt_c2_7__graph_2,dt_c1_7__graph_2])],[i1_7_2_1__graph_2,i2_7_2_1__graph_2,e1_7_2_1__graph_2]), [interesting(0.8),file(graph_2,e7_7__graph_2),[file(graph_2,e7_7__graph_2)]]). fof(t44_finseq_3,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(B,k2_finseq_1(A)) => k3_finseq_1(k14_finseq_1(B)) = k4_card_1(B) ) ) ) ), file(finseq_3,t44_finseq_3), [interesting(0.9),axiom,file(finseq_3,t44_finseq_3)]). fof(e11_7__graph_2,plain,( k3_finseq_1(k14_finseq_1(c6_7__graph_2)) = c1_7__graph_2 ), inference(mizar_by,[status(thm),assumptions([dt_c2_7__graph_2,dt_c1_7__graph_2])],[rc4_funct_1,existence_m1_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_finset_1,fc2_finseq_1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc6_finseq_1,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc3_nat_1,fc4_nat_1,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_nat_1,rc2_card_1,rc2_funct_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,t1_real,t2_subset,t4_real,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finseq_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc4_int_1,cc5_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc2_card_1,fc3_xreal_0,fc6_int_1,fc7_xreal_0,rc1_finseq_1,rc1_funct_1,rc1_int_1,rc1_xreal_0,rc2_int_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc6_arithm,t1_subset,t2_real,t3_real,t4_subset,t7_boole,spc1_numerals,spc1_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k4_card_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k2_finseq_1,dt_k3_finseq_1,dt_k4_card_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,t3_subset,t2_tarski,fraenkel_a_2_1_graph_2,e5_7__graph_2,e7_7__graph_2,t44_finseq_3]), [interesting(0.8),file(graph_2,e11_7__graph_2),[file(graph_2,e11_7__graph_2)]]). fof(s1_nat_1__e3_7_3__graph_2,theorem,( ! [A,B] : ( ( m2_subset_1(A,k1_numbers,k5_numbers) & m2_subset_1(B,k1_numbers,k5_numbers) ) => ( ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,0) & r1_xreal_0(0,A) & C = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),0) ) => r1_xreal_0(k1_nat_1(B,0),C) ) ) & ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ( ! [E] : ( m2_subset_1(E,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,A) & E = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),D) ) => r1_xreal_0(k1_nat_1(B,D),E) ) ) => ! [F] : ( m2_subset_1(F,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(D,1)) & r1_xreal_0(k1_nat_1(D,1),A) & F = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),k1_nat_1(D,1)) ) => r1_xreal_0(k1_nat_1(B,k1_nat_1(D,1)),F) ) ) ) ) ) => ! [D] : ( m2_subset_1(D,k1_numbers,k5_numbers) => ! [G] : ( m2_subset_1(G,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,D) & r1_xreal_0(D,A) & G = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),D) ) => r1_xreal_0(k1_nat_1(B,D),G) ) ) ) ) ) ), file(graph_2,s1_nat_1__e3_7_3__graph_2), [interesting(0.9),axiom,file(graph_2,s1_nat_1__e3_7_3__graph_2)]). fof(e1_7_3__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,0) & r1_xreal_0(0,c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),0) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,0),A) ) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,rqRealAdd__k2_xcmplx_0__r0_r0_r0,rqRealAdd__k2_xcmplx_0__r0_r1_r1,rqRealAdd__k2_xcmplx_0__r1_r0_r1,t1_numerals,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,t2_tarski,fraenkel_a_2_1_graph_2,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole]), [interesting(0.65),file(graph_2,e1_7_3__graph_2),[file(graph_2,e1_7_3__graph_2)]]). fof(dh_c1_7_3_1__graph_2,definition, ( ( m2_subset_1(c1_7_3_1__graph_2,k1_numbers,k5_numbers) => ( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,c1_7_3_1__graph_2) & r1_xreal_0(c1_7_3_1__graph_2,c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c1_7_3_1__graph_2) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2),A) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),A) ) ) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,c1_7__graph_2) & C = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),B) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,B),C) ) ) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(B,1)) & r1_xreal_0(k1_nat_1(B,1),c1_7__graph_2) & C = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(B,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(B,1)),C) ) ) ) ) ), introduced(definition,[new_symbol(c1_7_3_1__graph_2),file(graph_2,c1_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,c1_7_3_1__graph_2)]). fof(e1_7_3_1__graph_2,assumption,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,c1_7_3_1__graph_2) & r1_xreal_0(c1_7_3_1__graph_2,c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c1_7_3_1__graph_2) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2),A) ) ) ), introduced(assumption,[file(graph_2,e1_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,e1_7_3_1__graph_2)]). fof(dh_c2_7_3_1__graph_2,definition, ( ( m2_subset_1(c2_7_3_1__graph_2,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & c2_7_3_1__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),A) ) ) ), introduced(definition,[new_symbol(c2_7_3_1__graph_2),file(graph_2,c2_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,c2_7_3_1__graph_2)]). fof(e2_7_3_1__graph_2,assumption,( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) ), introduced(assumption,[file(graph_2,e2_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,e2_7_3_1__graph_2)]). fof(e3_7_3_1__graph_2,assumption,( r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) ), introduced(assumption,[file(graph_2,e3_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,e3_7_3_1__graph_2)]). fof(e4_7_3_1__graph_2,assumption,( c2_7_3_1__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ), introduced(assumption,[file(graph_2,e4_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,e4_7_3_1__graph_2)]). fof(e1_7_3_1_1_1__graph_2,assumption,( ~ r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),1) ), introduced(assumption,[file(graph_2,e1_7_3_1_1_1__graph_2)]), [interesting(0.2),axiom,file(graph_2,e1_7_3_1_1_1__graph_2)]). fof(dt_c1_7_3_1__graph_2,assumption,( m2_subset_1(c1_7_3_1__graph_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(graph_2,c1_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,c1_7_3_1__graph_2)]). fof(dt_c2_7_3_1__graph_2,assumption,( m2_subset_1(c2_7_3_1__graph_2,k1_numbers,k5_numbers) ), introduced(assumption,[file(graph_2,c2_7_3_1__graph_2)]), [interesting(0.5),axiom,file(graph_2,c2_7_3_1__graph_2)]). fof(de_c1_7_3_1_1_1__graph_2,definition,( c1_7_3_1_1_1__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c1_7_3_1__graph_2) ), introduced(definition,[new_symbol(c1_7_3_1_1_1__graph_2),file(graph_2,c1_7_3_1_1_1__graph_2)]), [interesting(0.2),axiom,file(graph_2,c1_7_3_1_1_1__graph_2)]). fof(e3_7_3_1_1_1__graph_2,plain,( m2_subset_1(k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c1_7_3_1__graph_2),k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_numerals,spc1_boole,existence_m2_subset_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2]), [interesting(0.2),file(graph_2,e3_7_3_1_1_1__graph_2),[file(graph_2,e3_7_3_1_1_1__graph_2)]]). fof(dt_c1_7_3_1_1_1__graph_2,plain,( m2_subset_1(c1_7_3_1_1_1__graph_2,k1_numbers,k5_numbers) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_boole,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,spc1_numerals,spc1_boole,existence_m2_subset_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,de_c1_7_3_1_1_1__graph_2,e3_7_3_1_1_1__graph_2]), [interesting(0.2),file(graph_2,c1_7_3_1_1_1__graph_2),[file(graph_2,c1_7_3_1_1_1__graph_2)]]). fof(e5_7_3_1__graph_2,plain,( ~ r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7_3_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_3_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c1_7_3_1__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,spc1_numerals,spc1_boole,t38_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(graph_2,e5_7_3_1__graph_2),[file(graph_2,e5_7_3_1__graph_2)]]). fof(e2_7_3_1_1_1__graph_2,plain, ( r1_xreal_0(1,c1_7_3_1__graph_2) & r1_xreal_0(c1_7_3_1__graph_2,c1_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,spc1_numerals,spc1_boole,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2,t38_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(graph_2,e2_7_3_1_1_1__graph_2),[file(graph_2,e2_7_3_1_1_1__graph_2)]]). fof(e5_7_3_1_1_1__graph_2,plain,( ~ r1_xreal_0(c2_7_3_1__graph_2,c1_7_3_1_1_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7_3_1__graph_2,dt_c2_7__graph_2,e4_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[rc4_funct_1,dt_k1_xboole_0,dt_k2_zfmisc_1,cc1_card_1,cc1_relset_1,fc14_finset_1,fc2_finseq_1,rc1_card_1,rc2_card_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_card_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t1_subset,t2_real,t2_subset,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k5_numbers,redefinition_m2_finseq_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_recdef_1,dt_k2_finseq_1,dt_k2_relat_1,dt_k3_finseq_1,dt_k5_numbers,dt_m2_finseq_1,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c1_7_3_1_1_1__graph_2,dt_c2_7__graph_2,dt_c2_7_3_1__graph_2,dt_c6_7__graph_2,de_c1_7_3_1_1_1__graph_2,de_c6_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,t3_subset,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e5_7__graph_2,e11_7__graph_2,e3_7_3_1__graph_2,e4_7_3_1__graph_2,e5_7_3_1__graph_2,e2_7_3_1_1_1__graph_2,d13_finseq_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(graph_2,e5_7_3_1_1_1__graph_2),[file(graph_2,e5_7_3_1_1_1__graph_2)]]). fof(e4_7_3_1_1_1__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2),c1_7_3_1_1_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7__graph_2,e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c1_7_3_1_1_1__graph_2,dt_c2_7__graph_2,de_c1_7_3_1_1_1__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e1_7_3_1__graph_2,e2_7_3_1_1_1__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(graph_2,e4_7_3_1_1_1__graph_2),[file(graph_2,e4_7_3_1_1_1__graph_2)]]). fof(e6_7_3_1_1_1__graph_2,plain,( ~ r1_xreal_0(c2_7_3_1__graph_2,k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c2_7__graph_2,e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finseq_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m2_finseq_1,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_subset_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_numbers,dt_k1_recdef_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,dt_c1_7__graph_2,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_c1_7_3_1__graph_2,dt_c1_7_3_1_1_1__graph_2,dt_c2_7__graph_2,dt_c2_7_3_1__graph_2,de_c1_7_3_1_1_1__graph_2,cc2_xreal_0,spc1_numerals,spc1_boole,e5_7_3_1_1_1__graph_2,e4_7_3_1_1_1__graph_2,t2_xreal_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(graph_2,e6_7_3_1_1_1__graph_2),[file(graph_2,e6_7_3_1_1_1__graph_2)]]). fof(e7_7_3_1_1_1__graph_2,plain,( r1_xreal_0(k1_nat_1(k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2),1),c2_7_3_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c2_7__graph_2,e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc2_finset_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc1_finset_1,rc2_card_1,rc2_nat_1,rc3_finset_1,rc3_nat_1,rc4_finset_1,t1_subset,t3_subset,t4_subset,t5_subset,existence_m1_subset_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k1_numbers,dt_k5_numbers,dt_m1_subset_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_int_1,rc2_xreal_0,rc3_xreal_0,rc4_xreal_0,spc6_arithm,t1_real,t2_real,t2_subset,t3_real,t4_real,t5_real,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_k2_xcmplx_0,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2,dt_c2_7_3_1__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,fc1_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_numerals,spc1_boole,e6_7_3_1_1_1__graph_2,t38_nat_1]), [interesting(0.2),file(graph_2,e7_7_3_1_1_1__graph_2),[file(graph_2,e7_7_3_1_1_1__graph_2)]]). fof(e8_7_3_1_1_1__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c2_7__graph_2,e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,spc6_arithm,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2,dt_c2_7_3_1__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1,spc1_numerals,spc1_boole,e7_7_3_1_1_1__graph_2]), [interesting(0.2),file(graph_2,e8_7_3_1_1_1__graph_2),[file(graph_2,e8_7_3_1_1_1__graph_2)]]). fof(i2_7_3_1_1_1__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_3_1_1_1__graph_2)]), [interesting(0.2),trivial,file(graph_2,i2_7_3_1_1_1__graph_2)]). fof(i1_7_3_1_1_1__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(conclusion,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c2_7__graph_2,e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_1__graph_2,e3_7_3_1__graph_2])],[e8_7_3_1_1_1__graph_2,i2_7_3_1_1_1__graph_2]), [interesting(0.2),file(graph_2,i1_7_3_1_1_1__graph_2),[file(graph_2,i1_7_3_1_1_1__graph_2)]]). fof(i1_7_3_1_1__graph_2,plain, ( ~ r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),1) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c2_7__graph_2,e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,e3_7_3_1__graph_2]),discharge_asm(discharge,[e1_7_3_1_1_1__graph_2])],[e1_7_3_1_1_1__graph_2,i1_7_3_1_1_1__graph_2]), [interesting(0.35),file(graph_2,i1_7_3_1_1__graph_2),[file(graph_2,i1_7_3_1_1__graph_2)]]). fof(e1_7_3_1_1_2__graph_2,assumption,( 1 = k1_nat_1(c1_7_3_1__graph_2,1) ), introduced(assumption,[file(graph_2,e1_7_3_1_1_2__graph_2)]), [interesting(0.2),axiom,file(graph_2,e1_7_3_1_1_2__graph_2)]). fof(dh_c1_7_3_1_1_2__graph_2,definition, ( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),1) & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),A) & r1_xreal_0(A,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) => ( m2_subset_1(c1_7_3_1_1_2__graph_2,k1_numbers,k5_numbers) & c1_7_3_1_1_2__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),1) & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),c1_7_3_1_1_2__graph_2) & r1_xreal_0(c1_7_3_1_1_2__graph_2,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) ), introduced(definition,[new_symbol(c1_7_3_1_1_2__graph_2),file(graph_2,c1_7_3_1_1_2__graph_2)]), [interesting(0.2),axiom,file(graph_2,c1_7_3_1_1_2__graph_2)]). fof(t27_finseq_3,theorem,( ! [A] : ( ( v1_relat_1(A) & v1_funct_1(A) & v1_finseq_1(A) ) => ! [B] : ( v4_ordinal2(B) => ( r2_hidden(B,k4_finseq_1(A)) <=> ( r1_xreal_0(1,B) & r1_xreal_0(B,k3_finseq_1(A)) ) ) ) ) ), file(finseq_3,t27_finseq_3), [interesting(0.9),axiom,file(finseq_3,t27_finseq_3)]). fof(e2_7_3_1_1_2__graph_2,plain,( r2_hidden(1,k4_finseq_1(k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,dt_c2_7__graph_2,dt_c1_7__graph_2,e3_7_3_1__graph_2])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_finset_1,rc2_finseq_1,rc4_funct_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc2_card_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc1_xreal_0,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_card_1,fc30_xreal_0,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_int_1,fc5_xreal_0,fc6_int_1,fc6_xreal_0,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,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,t1_real,t2_real,t2_subset,t3_real,t3_subset,t4_arithm,t4_real,t4_subset,t5_arithm,t5_real,t5_subset,t6_arithm,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k3_finseq_1,redefinition_k4_finseq_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k3_finseq_1,dt_k4_finseq_1,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,cc1_finseq_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,rc1_finseq_1,rc1_funct_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e1_7_3_1_1_2__graph_2,e11_7__graph_2,e3_7_3_1__graph_2,t27_finseq_3,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(graph_2,e2_7_3_1_1_2__graph_2),[file(graph_2,e2_7_3_1_1_2__graph_2)]]). fof(e3_7_3_1_1_2__graph_2,plain,( r2_hidden(k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),1),k2_relat_1(k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)))) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,dt_c2_7__graph_2,dt_c1_7__graph_2,e3_7_3_1__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc4_funct_1,commutativity_k2_xcmplx_0,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k2_xcmplx_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc2_xreal_0,cc3_int_1,cc3_nat_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_xreal_0,fc17_finseq_1,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_finset_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_finseq_1,rc4_finset_1,rc4_xreal_0,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t2_real,t3_real,t4_real,t5_real,t6_real,t7_real,t8_real,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,fc1_ordinal2,rc1_finseq_1,rc2_funct_1,rqLessOrEqual__r1_xreal_0__r1_r1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k1_recdef_1,redefinition_k4_finseq_1,redefinition_k5_numbers,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_recdef_1,dt_k1_relat_1,dt_k2_relat_1,dt_k4_finseq_1,dt_k5_numbers,dt_c1_7__graph_2,dt_c2_7__graph_2,rc1_funct_1,t1_subset,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e2_7_3_1_1_2__graph_2,d5_funct_1]), [interesting(0.2),file(graph_2,e3_7_3_1_1_2__graph_2),[file(graph_2,e3_7_3_1_1_2__graph_2)]]). fof(e12_7__graph_2,plain,( k2_relat_1(k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2))) = c6_7__graph_2 ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[rc4_funct_1,existence_m1_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_card_1,cc1_relset_1,fc14_finset_1,fc2_finseq_1,rc1_card_1,rc2_card_1,rc2_finseq_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_finseq_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_card_1,fc3_nat_1,fc3_xreal_0,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_xreal_0,fc9_xreal_0,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t1_real,t1_subset,t2_real,t2_subset,t3_real,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_boole,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k5_numbers,redefinition_m2_finseq_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k2_finseq_1,dt_k2_relat_1,dt_k3_finseq_1,dt_k5_numbers,dt_m2_finseq_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,rqLessOrEqual__r1_xreal_0__r1_r1,t3_subset,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e5_7__graph_2,d13_finseq_1]), [interesting(0.8),file(graph_2,e12_7__graph_2),[file(graph_2,e12_7__graph_2)]]). fof(e4_7_3_1_1_2__graph_2,plain,( ? [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),1) & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),A) & r1_xreal_0(A,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc4_funct_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_finseq_1,rc4_xreal_0,rc6_finseq_1,rc8_finseq_1,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc11_finseq_1,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,rc1_finset_1,rc1_funct_1,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k2_relat_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,t1_subset,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e3_7_3_1_1_2__graph_2,e12_7__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.2),file(graph_2,e4_7_3_1_1_2__graph_2),[file(graph_2,e4_7_3_1_1_2__graph_2)]]). fof(dt_c1_7_3_1_1_2__graph_2,plain,( m2_subset_1(c1_7_3_1_1_2__graph_2,k1_numbers,k5_numbers) ), inference(consider,[status(thm),assumptions([dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2])],[dh_c1_7_3_1_1_2__graph_2,e4_7_3_1_1_2__graph_2]), [interesting(0.2),file(graph_2,c1_7_3_1_1_2__graph_2),[file(graph_2,c1_7_3_1_1_2__graph_2)]]). fof(e5_7_3_1_1_2__graph_2,plain, ( c1_7_3_1_1_2__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),1) & r1_xreal_0(k1_nat_1(c2_7__graph_2,1),c1_7_3_1_1_2__graph_2) & r1_xreal_0(c1_7_3_1_1_2__graph_2,k1_nat_1(c2_7__graph_2,c1_7__graph_2)) ), inference(consider,[status(thm),assumptions([dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2])],[dh_c1_7_3_1_1_2__graph_2,e4_7_3_1_1_2__graph_2]), [interesting(0.2),file(graph_2,e5_7_3_1_1_2__graph_2),[file(graph_2,e5_7_3_1_1_2__graph_2)]]). fof(e6_7_3_1_1_2__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_funct_1,dt_k1_numbers,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_xreal_0,fc30_xreal_0,fc3_xreal_0,fc5_int_1,fc5_xreal_0,fc6_xreal_0,fc8_xreal_0,fc9_int_1,rc1_xreal_0,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,t1_real,t1_subset,t2_subset,t3_subset,t4_arithm,t4_real,t4_subset,t5_arithm,t5_subset,t6_arithm,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_recdef_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_7__graph_2,dt_c1_7_3_1__graph_2,dt_c1_7_3_1_1_2__graph_2,dt_c2_7__graph_2,dt_c2_7_3_1__graph_2,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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,t2_tarski,fraenkel_a_2_1_graph_2,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e4_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,e5_7_3_1_1_2__graph_2,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiv__k7_xcmplx_0__r1_rm1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0]), [interesting(0.2),file(graph_2,e6_7_3_1_1_2__graph_2),[file(graph_2,e6_7_3_1_1_2__graph_2)]]). fof(i2_7_3_1_1_2__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_3_1_1_2__graph_2)]), [interesting(0.2),trivial,file(graph_2,i2_7_3_1_1_2__graph_2)]). fof(i1_7_3_1_1_2__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(conclusion,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c1_7_3_1__graph_2,e1_7_3_1_1_2__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2])],[e6_7_3_1_1_2__graph_2,i2_7_3_1_1_2__graph_2]), [interesting(0.2),file(graph_2,i1_7_3_1_1_2__graph_2),[file(graph_2,i1_7_3_1_1_2__graph_2)]]). fof(i2_7_3_1_1__graph_2,plain, ( 1 = k1_nat_1(c1_7_3_1__graph_2,1) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,dt_c1_7_3_1__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[e1_7_3_1_1_2__graph_2])],[e1_7_3_1_1_2__graph_2,i1_7_3_1_1_2__graph_2]), [interesting(0.35),file(graph_2,i2_7_3_1_1__graph_2),[file(graph_2,i2_7_3_1_1__graph_2)]]). fof(d5_real_1,definition,( ! [A] : ( v1_xreal_0(A) => ! [B] : ( v1_xreal_0(B) => ( r1_xreal_0(A,B) <=> ~ ( r1_xreal_0(B,A) & B != A ) ) ) ) ), file(real_1,d5_real_1), [interesting(0.9),axiom,file(real_1,d5_real_1)]). fof(e1_7_3_1_1__graph_2,plain,( ~ ( r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),1) & 1 != k1_nat_1(c1_7_3_1__graph_2,1) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7_3_1__graph_2,e2_7_3_1__graph_2])],[reflexivity_r1_tarski,cc1_finseq_1,cc2_funct_1,rc1_finseq_1,rc1_funct_1,rc2_finset_1,rc2_funct_1,rc3_finseq_1,rc3_funct_1,rc4_funct_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,antisymmetry_r2_hidden,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_ordinal2,cc1_card_1,cc1_xreal_0,cc2_finset_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,t1_subset,t2_real,t3_real,t3_subset,t4_subset,t5_real,t5_subset,t6_real,t7_real,t8_real,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,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc3_xreal_0,fc8_xreal_0,rc1_xreal_0,spc6_arithm,t1_real,t2_subset,t4_real,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,dt_k1_nat_1,dt_c1_7_3_1__graph_2,cc2_xreal_0,spc1_numerals,spc1_boole,e2_7_3_1__graph_2,d5_real_1,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.35),file(graph_2,e1_7_3_1_1__graph_2),[file(graph_2,e1_7_3_1_1__graph_2)]]). fof(i4_7_3_1__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(percases,[status(thm),assumptions([e1_7_3_1__graph_2,dt_c2_7_3_1__graph_2,e4_7_3_1__graph_2,e3_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c1_7_3_1__graph_2,e2_7_3_1__graph_2])],[i1_7_3_1_1__graph_2,i2_7_3_1_1__graph_2,e1_7_3_1_1__graph_2]), [interesting(0.5),file(graph_2,i4_7_3_1__graph_2),[file(graph_2,i4_7_3_1__graph_2)]]). fof(i3_7_3_1__graph_2,plain, ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & c2_7_3_1__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ), inference(discharge_asm,[status(thm),assumptions([e1_7_3_1__graph_2,dt_c2_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c1_7_3_1__graph_2]),discharge_asm(discharge,[e2_7_3_1__graph_2,e3_7_3_1__graph_2,e4_7_3_1__graph_2])],[e2_7_3_1__graph_2,e3_7_3_1__graph_2,e4_7_3_1__graph_2,i4_7_3_1__graph_2]), [interesting(0.5),file(graph_2,i3_7_3_1__graph_2),[file(graph_2,i3_7_3_1__graph_2)]]). fof(i3_7_3_1_tmp__graph_2,plain, ( m2_subset_1(c2_7_3_1__graph_2,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & c2_7_3_1__graph_2 = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),c2_7_3_1__graph_2) ) ), inference(discharge_asm,[status(thm),assumptions([e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c1_7_3_1__graph_2]),discharge_asm(discharge,[dt_c2_7_3_1__graph_2])],[dt_c2_7_3_1__graph_2,i3_7_3_1__graph_2]), [interesting(0.5),i2_7_3_1__graph_2]). fof(i2_7_3_1__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),A) ) ) ), inference(let,[status(thm),assumptions([e1_7_3_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c1_7_3_1__graph_2])],[i3_7_3_1_tmp__graph_2,dh_c2_7_3_1__graph_2]), [interesting(0.5),file(graph_2,i2_7_3_1__graph_2),[file(graph_2,i2_7_3_1__graph_2)]]). fof(i1_7_3_1__graph_2,plain, ( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,c1_7_3_1__graph_2) & r1_xreal_0(c1_7_3_1__graph_2,c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c1_7_3_1__graph_2) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2),A) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),A) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c1_7_3_1__graph_2]),discharge_asm(discharge,[e1_7_3_1__graph_2])],[e1_7_3_1__graph_2,i2_7_3_1__graph_2]), [interesting(0.5),file(graph_2,i1_7_3_1__graph_2),[file(graph_2,i1_7_3_1__graph_2)]]). fof(i1_7_3_1_tmp__graph_2,plain, ( m2_subset_1(c1_7_3_1__graph_2,k1_numbers,k5_numbers) => ( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,c1_7_3_1__graph_2) & r1_xreal_0(c1_7_3_1__graph_2,c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c1_7_3_1__graph_2) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,c1_7_3_1__graph_2),A) ) ) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(c1_7_3_1__graph_2,1)) & r1_xreal_0(k1_nat_1(c1_7_3_1__graph_2,1),c1_7__graph_2) & A = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(c1_7_3_1__graph_2,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(c1_7_3_1__graph_2,1)),A) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[dt_c1_7_3_1__graph_2])],[dt_c1_7_3_1__graph_2,i1_7_3_1__graph_2]), [interesting(0.65),e2_7_3__graph_2]). fof(e2_7_3__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & B = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,A),B) ) ) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,k1_nat_1(A,1)) & r1_xreal_0(k1_nat_1(A,1),c1_7__graph_2) & B = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),k1_nat_1(A,1)) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,k1_nat_1(A,1)),B) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[i1_7_3_1_tmp__graph_2,dh_c1_7_3_1__graph_2]), [interesting(0.65),file(graph_2,e2_7_3__graph_2),[file(graph_2,e2_7_3__graph_2)]]). fof(e3_7_3__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & B = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,A),B) ) ) ) ), inference(mizar_from,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc6_finseq_1,redefinition_m2_relset_1,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,t2_tarski,fraenkel_a_2_1_graph_2,s1_nat_1__e3_7_3__graph_2,e1_7_3__graph_2,e2_7_3__graph_2]), [interesting(0.65),file(graph_2,e3_7_3__graph_2),[file(graph_2,e3_7_3__graph_2)]]). fof(i1_7_3__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i1_7_3__graph_2)]), [interesting(0.65),trivial,file(graph_2,i1_7_3__graph_2)]). fof(e18_7__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & B = k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) ) => r1_xreal_0(k1_nat_1(c2_7__graph_2,A),B) ) ) ) ), inference(conclusion,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2])],[e3_7_3__graph_2,i1_7_3__graph_2]), [interesting(0.8),file(graph_2,e18_7__graph_2),[file(graph_2,e18_7__graph_2)]]). fof(e19_7__graph_2,plain,( r1_xreal_0(k1_nat_1(c2_7__graph_2,c4_7__graph_2),c9_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c9_7__graph_2,de_c9_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e18_7__graph_2,e3_7__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.8),file(graph_2,e19_7__graph_2),[file(graph_2,e19_7__graph_2)]]). fof(t45_finseq_3,theorem,( ! [A] : ( v4_ordinal2(A) => ! [B] : ( v1_finset_1(B) => ( r1_tarski(B,k2_finseq_1(A)) => k4_finseq_1(k14_finseq_1(B)) = k2_finseq_1(k4_card_1(B)) ) ) ) ), file(finseq_3,t45_finseq_3), [interesting(0.9),axiom,file(finseq_3,t45_finseq_3)]). fof(e10_7__graph_2,plain,( k4_finseq_1(k14_finseq_1(c6_7__graph_2)) = k2_finseq_1(c1_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c2_7__graph_2,dt_c1_7__graph_2])],[rc4_funct_1,existence_m1_relset_1,dt_k1_xboole_0,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_finset_1,fc2_finseq_1,rc2_finseq_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc6_finseq_1,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finset_1,cc1_funct_1,cc2_funct_1,cc3_xreal_0,cc4_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_ordinal2,fc3_nat_1,fc4_nat_1,fc8_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finset_1,rc1_nat_1,rc2_card_1,rc2_funct_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,t1_real,t2_subset,t4_real,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k2_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finseq_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc4_int_1,cc5_xreal_0,fc17_finseq_1,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc2_card_1,fc3_xreal_0,fc6_int_1,fc7_xreal_0,rc1_finseq_1,rc1_funct_1,rc1_int_1,rc1_xreal_0,rc2_int_1,rqLessOrEqual__r1_xreal_0__r1_r1,spc6_arithm,t1_subset,t2_real,t3_real,t4_subset,t7_boole,spc1_numerals,spc1_boole,commutativity_k1_nat_1,reflexivity_r1_tarski,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k4_card_1,redefinition_k4_finseq_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k2_finseq_1,dt_k4_card_1,dt_k4_finseq_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c6_7__graph_2,de_c6_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,t3_subset,t2_tarski,fraenkel_a_2_1_graph_2,e5_7__graph_2,e7_7__graph_2,t45_finseq_3]), [interesting(0.8),file(graph_2,e10_7__graph_2),[file(graph_2,e10_7__graph_2)]]). fof(e20_7__graph_2,plain, ( r1_xreal_0(1,c8_7__graph_2) & r1_xreal_0(c8_7__graph_2,c1_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[existence_m1_relset_1,dt_k2_zfmisc_1,dt_m1_relset_1,cc1_relset_1,fc14_finset_1,rc2_finseq_1,rc4_funct_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,fc1_ordinal2,fc2_finseq_1,rc1_card_1,rc2_card_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,commutativity_k2_xcmplx_0,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_k5_numbers,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_numbers,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc1_xreal_0,fc20_xreal_0,fc27_xreal_0,fc28_xreal_0,fc29_xreal_0,fc30_xreal_0,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_int_1,fc5_xreal_0,fc6_int_1,fc6_xreal_0,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,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,t1_real,t2_real,t2_subset,t3_real,t3_subset,t4_arithm,t4_real,t4_subset,t5_arithm,t5_real,t5_subset,t6_arithm,t6_boole,t6_real,t7_real,t8_boole,t8_real,t2_tarski,fraenkel_a_2_1_graph_2,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k2_finseq_1,redefinition_k4_finseq_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k2_finseq_1,dt_k4_finseq_1,dt_k4_xcmplx_0,dt_k6_xcmplx_0,dt_k7_xcmplx_0,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,dt_c7_7__graph_2,dt_c8_7__graph_2,de_c6_7__graph_2,de_c8_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_rm1,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,t7_boole,spc0_numerals,spc1_numerals,spc0_boole,spc1_boole,e10_7__graph_2,e15_7__graph_2,t3_finseq_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(0.8),file(graph_2,e20_7__graph_2),[file(graph_2,e20_7__graph_2)]]). fof(e2_7_4_1__graph_2,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([e1_7_4_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[rc4_funct_1,dt_k1_xboole_0,dt_k2_zfmisc_1,cc1_card_1,cc1_relset_1,fc14_finset_1,fc2_finseq_1,rc1_card_1,rc2_card_1,rc2_finset_1,rc2_nat_1,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc4_finseq_1,rc6_finseq_1,commutativity_k2_xcmplx_0,existence_m1_finseq_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,existence_m2_subset_1,redefinition_m2_relset_1,redefinition_m2_subset_1,dt_k1_card_1,dt_k1_finseq_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_finseq_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc6_xreal_0,cc7_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_finseq_1,fc11_xreal_0,fc12_xreal_0,fc13_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc19_xreal_0,fc1_finseq_1,fc1_int_1,fc1_nat_1,fc1_ordinal2,fc1_xreal_0,fc20_xreal_0,fc2_card_1,fc3_int_1,fc3_nat_1,fc3_xreal_0,fc4_int_1,fc4_nat_1,fc5_int_1,fc5_xreal_0,fc6_int_1,fc7_xreal_0,fc8_int_1,fc8_xreal_0,fc9_int_1,fc9_xreal_0,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_xreal_0,rc3_finset_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,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,t1_real,t2_real,t2_subset,t3_real,t4_arithm,t4_real,t4_subset,t5_real,t5_subset,t6_boole,t6_real,t7_real,t8_boole,t8_real,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_tarski,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,existence_m2_finseq_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k2_finseq_1,redefinition_k3_finseq_1,redefinition_k4_finseq_1,redefinition_k5_numbers,redefinition_m2_finseq_1,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_recdef_1,dt_k2_finseq_1,dt_k2_relat_1,dt_k3_finseq_1,dt_k4_finseq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_m2_finseq_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,dt_c7_7__graph_2,dt_c8_7__graph_2,dt_c9_7__graph_2,de_c6_7__graph_2,de_c8_7__graph_2,de_c9_7__graph_2,cc1_xreal_0,cc3_int_1,cc3_nat_1,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t3_subset,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e1_7_4_1__graph_2,e3_7__graph_2,e5_7__graph_2,e11_7__graph_2,e15_7__graph_2,e19_7__graph_2,e20_7__graph_2,d13_finseq_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(graph_2,e2_7_4_1__graph_2),[file(graph_2,e2_7_4_1__graph_2)]]). fof(i2_7_4_1__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_4_1__graph_2)]), [interesting(0.5),trivial,file(graph_2,i2_7_4_1__graph_2)]). fof(i1_7_4_1__graph_2,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([e1_7_4_1__graph_2,dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[e2_7_4_1__graph_2,i2_7_4_1__graph_2]), [interesting(0.5),file(graph_2,i1_7_4_1__graph_2),[file(graph_2,i1_7_4_1__graph_2)]]). fof(i1_7_4__graph_2,plain,( r1_xreal_0(c8_7__graph_2,c4_7__graph_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2]),discharge_asm(discharge,[e1_7_4_1__graph_2])],[e1_7_4_1__graph_2,i1_7_4_1__graph_2]), [interesting(0.65),file(graph_2,i1_7_4__graph_2),[file(graph_2,i1_7_4__graph_2)]]). fof(e1_7_4_2__graph_2,assumption,( ~ r1_xreal_0(c4_7__graph_2,c8_7__graph_2) ), introduced(assumption,[file(graph_2,e1_7_4_2__graph_2)]), [interesting(0.5),axiom,file(graph_2,e1_7_4_2__graph_2)]). fof(t2_xcmplx_1,theorem,( ! [A] : ( v1_xcmplx_0(A) => ! [B] : ( v1_xcmplx_0(B) => ! [C] : ( v1_xcmplx_0(C) => ( k2_xcmplx_0(A,B) = k2_xcmplx_0(C,B) => A = C ) ) ) ) ), file(xcmplx_1,t2_xcmplx_1), [interesting(0.9),axiom,file(xcmplx_1,t2_xcmplx_1)]). fof(e2_7_4_2__graph_2,plain,( k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),c8_7__graph_2) != k1_nat_1(c2_7__graph_2,c8_7__graph_2) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc4_funct_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_finseq_1,rc4_xreal_0,rc6_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc1_xreal_0,fc3_xreal_0,fc5_int_1,fc5_xreal_0,fc8_xreal_0,fc9_int_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_xreal_0,rc2_finseq_1,rc2_funct_1,rc3_finset_1,rc4_finset_1,rc7_finseq_1,rc8_finseq_1,t1_numerals,t1_real,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,commutativity_k2_xcmplx_0,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k4_finseq_1,redefinition_k5_numbers,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_recdef_1,dt_k2_xcmplx_0,dt_k4_finseq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,dt_c7_7__graph_2,dt_c8_7__graph_2,de_c6_7__graph_2,de_c8_7__graph_2,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r1,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,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,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,rqRealDiff__k6_xcmplx_0__r1_r2_rm1,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,spc1_arithm,spc6_arithm,spc8_arithm,spc9_arithm,t1_arithm,t1_subset,t4_arithm,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_7__graph_2,e15_7__graph_2,t2_xcmplx_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqRealDiff__k6_xcmplx_0__r1_rm1_r2]), [interesting(0.5),file(graph_2,e2_7_4_2__graph_2),[file(graph_2,e2_7_4_2__graph_2)]]). fof(e4_7__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) & k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) != k1_nat_1(c2_7__graph_2,A) ) => r1_xreal_0(c4_7__graph_2,A) ) ) ), inference(consider,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dh_c4_7__graph_2,e2_7__graph_2]), [interesting(0.8),file(graph_2,e4_7__graph_2),[file(graph_2,e4_7__graph_2)]]). fof(e3_7_4_2__graph_2,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2,e1_7_4_2__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc2_finset_1,rc3_finseq_1,rc3_funct_1,rc4_finseq_1,rc4_funct_1,rc6_finseq_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_finseq_1,cc1_xreal_0,cc2_finset_1,cc2_funct_1,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc1_int_1,fc1_nat_1,fc2_finseq_1,fc3_nat_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc9_xreal_0,rc1_card_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_int_1,rc1_nat_1,rc1_xreal_0,rc2_card_1,rc2_finseq_1,rc2_funct_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finset_1,rc3_nat_1,rc3_xreal_0,rc4_finset_1,rc4_xreal_0,rc7_finseq_1,rc8_finseq_1,spc6_arithm,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,antisymmetry_r2_hidden,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,redefinition_m2_finseq_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_c7_7__graph_2,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_int_1,cc2_nat_1,cc2_xreal_0,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc1_ordinal2,fc3_xreal_0,fc8_xreal_0,t1_real,t1_subset,t2_subset,t3_subset,t4_real,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole,commutativity_k1_nat_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,existence_m2_subset_1,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k5_numbers,redefinition_m2_subset_1,dt_k14_finseq_1,dt_k1_nat_1,dt_k1_numbers,dt_k1_recdef_1,dt_k5_numbers,dt_m2_subset_1,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c8_7__graph_2,de_c8_7__graph_2,t2_tarski,fraenkel_a_2_1_graph_2,spc1_numerals,spc1_boole,e2_7_4_2__graph_2,e4_7__graph_2,e20_7__graph_2,e1_7_4_2__graph_2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.5),file(graph_2,e3_7_4_2__graph_2),[file(graph_2,e3_7_4_2__graph_2)]]). fof(i2_7_4_2__graph_2,theorem,( $true ), introduced(tautology,[file(graph_2,i2_7_4_2__graph_2)]), [interesting(0.5),trivial,file(graph_2,i2_7_4_2__graph_2)]). fof(i1_7_4_2__graph_2,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2,e1_7_4_2__graph_2])],[e3_7_4_2__graph_2,i2_7_4_2__graph_2]), [interesting(0.5),file(graph_2,i1_7_4_2__graph_2),[file(graph_2,i1_7_4_2__graph_2)]]). fof(i2_7_4__graph_2,plain,( r1_xreal_0(c4_7__graph_2,c8_7__graph_2) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2]),discharge_asm(discharge,[e1_7_4_2__graph_2])],[e1_7_4_2__graph_2,i1_7_4_2__graph_2]), [interesting(0.65),file(graph_2,i2_7_4__graph_2),[file(graph_2,i2_7_4__graph_2)]]). fof(e1_7_4__graph_2,plain,( ~ ( r1_xreal_0(c8_7__graph_2,c4_7__graph_2) & r1_xreal_0(c4_7__graph_2,c8_7__graph_2) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[dt_k2_zfmisc_1,cc1_relset_1,fc14_finset_1,rc4_funct_1,reflexivity_r1_tarski,existence_m1_finseq_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_xboole_0,dt_m1_finseq_1,dt_m2_relset_1,cc1_card_1,cc1_xreal_0,cc3_int_1,cc3_nat_1,cc4_int_1,cc4_xreal_0,cc5_xreal_0,cc8_xreal_0,fc10_xreal_0,fc11_xreal_0,fc12_xreal_0,fc14_xreal_0,fc15_xreal_0,fc16_xreal_0,fc19_xreal_0,fc1_int_1,fc1_nat_1,fc20_xreal_0,fc2_finseq_1,fc3_int_1,fc3_nat_1,fc4_int_1,fc4_nat_1,fc6_int_1,fc7_xreal_0,fc8_int_1,fc9_xreal_0,rc1_card_1,rc1_int_1,rc1_nat_1,rc2_card_1,rc2_finset_1,rc2_int_1,rc2_nat_1,rc2_xreal_0,rc3_finseq_1,rc3_funct_1,rc3_nat_1,rc3_xreal_0,rc4_finseq_1,rc4_xreal_0,rc6_finseq_1,t2_real,t3_real,t5_real,t6_real,t7_real,t8_real,commutativity_k2_xcmplx_0,existence_m1_relset_1,existence_m1_subset_1,existence_m2_finseq_1,existence_m2_subset_1,redefinition_m2_finseq_1,redefinition_m2_subset_1,dt_k1_numbers,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xcmplx_0,dt_k5_ordinal2,dt_m1_relset_1,dt_m1_subset_1,dt_m2_finseq_1,dt_m2_subset_1,cc1_finseq_1,cc1_finset_1,cc1_funct_1,cc1_nat_1,cc2_card_1,cc2_finset_1,cc2_funct_1,cc2_int_1,cc2_nat_1,cc3_card_1,cc3_xreal_0,cc6_xreal_0,cc7_xreal_0,fc13_xreal_0,fc17_finseq_1,fc17_xreal_0,fc18_xreal_0,fc1_ordinal2,fc3_xreal_0,fc5_int_1,fc8_xreal_0,fc9_int_1,rc1_finseq_1,rc1_finset_1,rc1_funct_1,rc1_xreal_0,rc2_finseq_1,rc2_funct_1,rc3_finset_1,rc4_finset_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,t1_real,t2_subset,t3_subset,t4_arithm,t4_real,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k1_nat_1,involutiveness_k4_xcmplx_0,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,antisymmetry_r2_hidden,redefinition_k1_nat_1,redefinition_k1_recdef_1,redefinition_k4_finseq_1,redefinition_k5_numbers,dt_k14_finseq_1,dt_k1_funct_1,dt_k1_nat_1,dt_k1_recdef_1,dt_k4_finseq_1,dt_k4_xcmplx_0,dt_k5_numbers,dt_k6_xcmplx_0,dt_c1_7__graph_2,dt_c2_7__graph_2,dt_c4_7__graph_2,dt_c6_7__graph_2,dt_c7_7__graph_2,dt_c8_7__graph_2,de_c6_7__graph_2,de_c8_7__graph_2,cc2_xreal_0,fc1_xreal_0,fc5_xreal_0,rqLessOrEqual__r1_xreal_0__r0_r0,rqLessOrEqual__r1_xreal_0__r0_r1,rqLessOrEqual__r1_xreal_0__r0_r2,rqLessOrEqual__r1_xreal_0__r0_rm1,rqLessOrEqual__r1_xreal_0__r0_rm2,rqLessOrEqual__r1_xreal_0__r1_r0,rqLessOrEqual__r1_xreal_0__r1_r2,rqLessOrEqual__r1_xreal_0__r1_rm1,rqLessOrEqual__r1_xreal_0__r1_rm2,rqLessOrEqual__r1_xreal_0__r2_r0,rqLessOrEqual__r1_xreal_0__r2_r1,rqLessOrEqual__r1_xreal_0__r2_r2,rqLessOrEqual__r1_xreal_0__r2_rm1,rqLessOrEqual__r1_xreal_0__r2_rm2,rqLessOrEqual__r1_xreal_0__rm1_r0,rqLessOrEqual__r1_xreal_0__rm1_r1,rqLessOrEqual__r1_xreal_0__rm1_r2,rqLessOrEqual__r1_xreal_0__rm1_rm1,rqLessOrEqual__r1_xreal_0__rm1_rm2,rqLessOrEqual__r1_xreal_0__rm2_r0,rqLessOrEqual__r1_xreal_0__rm2_r1,rqLessOrEqual__r1_xreal_0__rm2_r2,rqLessOrEqual__r1_xreal_0__rm2_rm1,rqLessOrEqual__r1_xreal_0__rm2_rm2,rqRealDiff__k6_xcmplx_0__r0_r0_r0,rqRealDiff__k6_xcmplx_0__r0_r1_rm1,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_r0_r1,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__r0_r0,rqRealNeg__k4_xcmplx_0__r2_rm2,rqRealNeg__k4_xcmplx_0__rm2_r2,t1_subset,t7_boole,t2_tarski,fraenkel_a_2_1_graph_2,spc0_numerals,spc1_numerals,spc2_numerals,spc0_boole,spc1_boole,spc2_boole,e3_7__graph_2,e15_7__graph_2,d5_real_1,rqRealNeg__k4_xcmplx_0__r1_rm1,rqRealDiff__k6_xcmplx_0__r1_r1_r0,rqRealNeg__k4_xcmplx_0__rm1_r1,rqRealDiff__k6_xcmplx_0__rm1_rm1_r0,rqRealDiff__k6_xcmplx_0__rm1_r1_rm2,rqLessOrEqual__r1_xreal_0__r1_r1]), [interesting(0.65),file(graph_2,e1_7_4__graph_2),[file(graph_2,e1_7_4__graph_2)]]). fof(i3_7__graph_2,plain,( ~ $true ), inference(percases,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2,e1_7__graph_2])],[i1_7_4__graph_2,i2_7_4__graph_2,e1_7_4__graph_2]), [interesting(0.8),file(graph_2,i3_7__graph_2),[file(graph_2,i3_7__graph_2)]]). fof(i2_7__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) = k1_nat_1(c2_7__graph_2,A) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2,dt_c2_7__graph_2]),discharge_asm(discharge,[e1_7__graph_2])],[e1_7__graph_2,i3_7__graph_2]), [interesting(0.8),file(graph_2,i2_7__graph_2),[file(graph_2,i2_7__graph_2)]]). fof(i2_7_tmp__graph_2,plain, ( m2_subset_1(c2_7__graph_2,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,A) & r1_xreal_0(A,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,c2_7__graph_2)),A) = k1_nat_1(c2_7__graph_2,A) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c1_7__graph_2]),discharge_asm(discharge,[dt_c2_7__graph_2])],[dt_c2_7__graph_2,i2_7__graph_2]), [interesting(0.8),i1_7__graph_2]). fof(i1_7__graph_2,plain,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,A)),B) = k1_nat_1(A,B) ) ) ) ), inference(let,[status(thm),assumptions([dt_c1_7__graph_2])],[i2_7_tmp__graph_2,dh_c2_7__graph_2]), [interesting(0.8),file(graph_2,i1_7__graph_2),[file(graph_2,i1_7__graph_2)]]). fof(i1_7_tmp__graph_2,plain, ( m2_subset_1(c1_7__graph_2,k1_numbers,k5_numbers) => ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,B) & r1_xreal_0(B,c1_7__graph_2) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(c1_7__graph_2,A)),B) = k1_nat_1(A,B) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_7__graph_2])],[dt_c1_7__graph_2,i1_7__graph_2]), [interesting(1),t5_graph_2]). fof(t5_graph_2,theorem,( ! [A] : ( m2_subset_1(A,k1_numbers,k5_numbers) => ! [B] : ( m2_subset_1(B,k1_numbers,k5_numbers) => ! [C] : ( m2_subset_1(C,k1_numbers,k5_numbers) => ( ( r1_xreal_0(1,C) & r1_xreal_0(C,A) ) => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),C) = k1_nat_1(B,C) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_7_tmp__graph_2,dh_c1_7__graph_2]), [interesting(1),file(graph_2,t5_graph_2),[file(graph_2,t5_graph_2)]]).