% Mizar ND problem: t2_boolmark,boolmark,58,16 fof(dh_c1_2__boolmark,definition, ( ( ~ v1_xboole_0(c1_2__boolmark) => ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(c1_2__boolmark)) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(c1_2__boolmark)) => ! [D] : ( ( v1_funct_1(D) & v1_funct_2(D,c1_2__boolmark,A) & m2_relset_1(D,c1_2__boolmark,A) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,A,D,B),k2_funct_2(c1_2__boolmark,A,D,C)) => r1_xboole_0(B,C) ) ) ) ) ) ) => ! [E] : ( ~ v1_xboole_0(E) => ! [F] : ( ~ v1_xboole_0(F) => ! [G] : ( m1_subset_1(G,k1_zfmisc_1(E)) => ! [H] : ( m1_subset_1(H,k1_zfmisc_1(E)) => ! [I] : ( ( v1_funct_1(I) & v1_funct_2(I,E,F) & m2_relset_1(I,E,F) ) => ( r1_xboole_0(k2_funct_2(E,F,I,G),k2_funct_2(E,F,I,H)) => r1_xboole_0(G,H) ) ) ) ) ) ) ), introduced(definition,[new_symbol(c1_2__boolmark),file(boolmark,c1_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c1_2__boolmark)]). fof(dh_c2_2__boolmark,definition, ( ( ~ v1_xboole_0(c2_2__boolmark) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(c1_2__boolmark)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(c1_2__boolmark)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(C,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,A),k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,B)) => r1_xboole_0(A,B) ) ) ) ) ) => ! [D] : ( ~ v1_xboole_0(D) => ! [E] : ( m1_subset_1(E,k1_zfmisc_1(c1_2__boolmark)) => ! [F] : ( m1_subset_1(F,k1_zfmisc_1(c1_2__boolmark)) => ! [G] : ( ( v1_funct_1(G) & v1_funct_2(G,c1_2__boolmark,D) & m2_relset_1(G,c1_2__boolmark,D) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,D,G,E),k2_funct_2(c1_2__boolmark,D,G,F)) => r1_xboole_0(E,F) ) ) ) ) ) ), introduced(definition,[new_symbol(c2_2__boolmark),file(boolmark,c2_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c2_2__boolmark)]). fof(dh_c3_2__boolmark,definition, ( ( m1_subset_1(c3_2__boolmark,k1_zfmisc_1(c1_2__boolmark)) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(c1_2__boolmark)) => ! [B] : ( ( v1_funct_1(B) & v1_funct_2(B,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(B,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,B,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,B,A)) => r1_xboole_0(c3_2__boolmark,A) ) ) ) ) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(c1_2__boolmark)) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(c1_2__boolmark)) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(E,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,E,C),k2_funct_2(c1_2__boolmark,c2_2__boolmark,E,D)) => r1_xboole_0(C,D) ) ) ) ) ), introduced(definition,[new_symbol(c3_2__boolmark),file(boolmark,c3_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c3_2__boolmark)]). fof(dh_c4_2__boolmark,definition, ( ( m1_subset_1(c4_2__boolmark,k1_zfmisc_1(c1_2__boolmark)) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(A,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ) ) ) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(c1_2__boolmark)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(C,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,B)) => r1_xboole_0(c3_2__boolmark,B) ) ) ) ), introduced(definition,[new_symbol(c4_2__boolmark),file(boolmark,c4_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c4_2__boolmark)]). fof(dh_c5_2__boolmark,definition, ( ( ( v1_funct_1(c5_2__boolmark) & v1_funct_2(c5_2__boolmark,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(c5_2__boolmark,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ) ) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(A,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ) ) ), introduced(definition,[new_symbol(c5_2__boolmark),file(boolmark,c5_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c5_2__boolmark)]). 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_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(fc4_subset_1,theorem,( ! [A,B] : ( ( ~ v1_xboole_0(A) & ~ v1_xboole_0(B) ) => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ), file(subset_1,fc4_subset_1), [interesting(0.9),axiom,file(subset_1,fc4_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(dt_k1_zfmisc_1,axiom,( $true ), file(zfmisc_1,k1_zfmisc_1), [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_1)]). fof(dt_k9_relat_1,axiom,( $true ), file(relat_1,k9_relat_1), [interesting(0.9),axiom,file(relat_1,k9_relat_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(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(fc1_subset_1,theorem,( ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ), file(subset_1,fc1_subset_1), [interesting(0.9),axiom,file(subset_1,fc1_subset_1)]). fof(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_margrel1,theorem,( ? [A] : v1_margrel1(A) ), file(margrel1,rc1_margrel1), [interesting(0.9),axiom,file(margrel1,rc1_margrel1)]). fof(rc1_subset_1,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & ~ v1_xboole_0(B) ) ) ), file(subset_1,rc1_subset_1), [interesting(0.9),axiom,file(subset_1,rc1_subset_1)]). fof(rc2_subset_1,theorem,( ! [A] : ? [B] : ( m1_subset_1(B,k1_zfmisc_1(A)) & v1_xboole_0(B) ) ), file(subset_1,rc2_subset_1), [interesting(0.9),axiom,file(subset_1,rc2_subset_1)]). fof(rc3_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) & v1_finseq_1(A) ) ), file(finseq_1,rc3_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc3_finseq_1)]). fof(rc6_finseq_1,theorem,( ? [A] : ( v1_xboole_0(A) & v1_relat_1(A) & v1_funct_1(A) & v2_funct_1(A) & v1_finset_1(A) ) ), file(finseq_1,rc6_finseq_1), [interesting(0.9),axiom,file(finseq_1,rc6_finseq_1)]). fof(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(commutativity_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(idempotence_k3_xboole_0,theorem,( ! [A,B] : k3_xboole_0(A,A) = A ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(commutativity_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k5_subset_1(A,C,B) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(idempotence_k5_subset_1,theorem,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,B) = B ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(symmetry_r1_xboole_0,theorem,( ! [A,B] : ( r1_xboole_0(A,B) => r1_xboole_0(B,A) ) ), file(xboole_0,r1_xboole_0), [interesting(0.9),axiom,file(xboole_0,r1_xboole_0)]). fof(redefinition_k2_funct_2,definition,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => k2_funct_2(A,B,C,D) = k9_relat_1(C,D) ) ), file(funct_2,k2_funct_2), [interesting(0.9),axiom,file(funct_2,k2_funct_2)]). fof(redefinition_k5_subset_1,definition,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => k5_subset_1(A,B,C) = k3_xboole_0(B,C) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(dt_k1_xboole_0,axiom,( $true ), file(xboole_0,k1_xboole_0), [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]). fof(dt_k2_funct_2,axiom,( ! [A,B,C,D] : ( ( v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) ) => m1_subset_1(k2_funct_2(A,B,C,D),k1_zfmisc_1(B)) ) ), file(funct_2,k2_funct_2), [interesting(0.9),axiom,file(funct_2,k2_funct_2)]). fof(dt_k3_xboole_0,axiom,( $true ), file(xboole_0,k3_xboole_0), [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]). fof(dt_k5_subset_1,axiom,( ! [A,B,C] : ( ( m1_subset_1(B,k1_zfmisc_1(A)) & m1_subset_1(C,k1_zfmisc_1(A)) ) => m1_subset_1(k5_subset_1(A,B,C),k1_zfmisc_1(A)) ) ), file(subset_1,k5_subset_1), [interesting(0.9),axiom,file(subset_1,k5_subset_1)]). fof(dt_c1_2__boolmark,assumption,( ~ v1_xboole_0(c1_2__boolmark) ), introduced(assumption,[file(boolmark,c1_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c1_2__boolmark)]). fof(dt_c2_2__boolmark,assumption,( ~ v1_xboole_0(c2_2__boolmark) ), introduced(assumption,[file(boolmark,c2_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c2_2__boolmark)]). fof(dt_c3_2__boolmark,assumption,( m1_subset_1(c3_2__boolmark,k1_zfmisc_1(c1_2__boolmark)) ), introduced(assumption,[file(boolmark,c3_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c3_2__boolmark)]). fof(dt_c4_2__boolmark,assumption,( m1_subset_1(c4_2__boolmark,k1_zfmisc_1(c1_2__boolmark)) ), introduced(assumption,[file(boolmark,c4_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c4_2__boolmark)]). fof(dt_c5_2__boolmark,assumption, ( v1_funct_1(c5_2__boolmark) & v1_funct_2(c5_2__boolmark,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(c5_2__boolmark,c1_2__boolmark,c2_2__boolmark) ), introduced(assumption,[file(boolmark,c5_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c5_2__boolmark)]). fof(fc1_margrel1,theorem, ( v1_xboole_0(k1_xboole_0) & v1_margrel1(k1_xboole_0) ), file(margrel1,fc1_margrel1), [interesting(0.9),axiom,file(margrel1,fc1_margrel1)]). 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(e1_2__boolmark,assumption,( k5_subset_1(c2_2__boolmark,k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c4_2__boolmark)) = k1_xboole_0 ), introduced(assumption,[file(boolmark,e1_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,e1_2__boolmark)]). fof(d7_xboole_0,definition,( ! [A,B] : ( r1_xboole_0(A,B) <=> k3_xboole_0(A,B) = k1_xboole_0 ) ), file(xboole_0,d7_xboole_0), [interesting(0.9),axiom,file(xboole_0,d7_xboole_0)]). fof(e2_2__boolmark,assumption,( k5_subset_1(c1_2__boolmark,c3_2__boolmark,c4_2__boolmark) != k1_xboole_0 ), introduced(assumption,[file(boolmark,e2_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,e2_2__boolmark)]). fof(reflexivity_r1_tarski,theorem,( ! [A,B] : r1_tarski(A,A) ), file(tarski,r1_tarski), [interesting(0.9),axiom,file(tarski,r1_tarski)]). 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(dt_k1_funct_1,axiom,( $true ), file(funct_1,k1_funct_1), [interesting(0.9),axiom,file(funct_1,k1_funct_1)]). fof(t2_subset,theorem,( ! [A,B] : ( m1_subset_1(A,B) => ( v1_xboole_0(B) | r2_hidden(A,B) ) ) ), file(subset,t2_subset), [interesting(0.9),axiom,file(subset,t2_subset)]). fof(t3_subset,theorem,( ! [A,B] : ( m1_subset_1(A,k1_zfmisc_1(B)) <=> r1_tarski(A,B) ) ), file(subset,t3_subset), [interesting(0.9),axiom,file(subset,t3_subset)]). fof(t4_subset,theorem,( ! [A,B,C] : ( ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) ) => m1_subset_1(A,C) ) ), file(subset,t4_subset), [interesting(0.9),axiom,file(subset,t4_subset)]). fof(t5_subset,theorem,( ! [A,B,C] : ~ ( r2_hidden(A,B) & m1_subset_1(B,k1_zfmisc_1(C)) & v1_xboole_0(C) ) ), file(subset,t5_subset), [interesting(0.9),axiom,file(subset,t5_subset)]). fof(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(antisymmetry_r2_hidden,theorem,( ! [A,B] : ( r2_hidden(A,B) => ~ r2_hidden(B,A) ) ), file(hidden,r2_hidden), [interesting(0.9),axiom,file(hidden,r2_hidden)]). fof(redefinition_k8_funct_2,definition,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => k8_funct_2(A,B,C,D) = k1_funct_1(C,D) ) ), file(funct_2,k8_funct_2), [interesting(0.9),axiom,file(funct_2,k8_funct_2)]). fof(dt_k8_funct_2,axiom,( ! [A,B,C,D] : ( ( ~ v1_xboole_0(A) & v1_funct_1(C) & v1_funct_2(C,A,B) & m1_relset_1(C,A,B) & m1_subset_1(D,A) ) => m1_subset_1(k8_funct_2(A,B,C,D),B) ) ), file(funct_2,k8_funct_2), [interesting(0.9),axiom,file(funct_2,k8_funct_2)]). fof(dh_c6_2__boolmark,definition, ( ? [A] : ( m1_subset_1(A,c1_2__boolmark) & r2_hidden(A,k5_subset_1(c1_2__boolmark,c3_2__boolmark,c4_2__boolmark)) ) => ( m1_subset_1(c6_2__boolmark,c1_2__boolmark) & r2_hidden(c6_2__boolmark,k5_subset_1(c1_2__boolmark,c3_2__boolmark,c4_2__boolmark)) ) ), introduced(definition,[new_symbol(c6_2__boolmark),file(boolmark,c6_2__boolmark)]), [interesting(0.8),axiom,file(boolmark,c6_2__boolmark)]). fof(t2_boole,theorem,( ! [A] : k3_xboole_0(A,k1_xboole_0) = k1_xboole_0 ), file(boole,t2_boole), [interesting(0.9),axiom,file(boole,t2_boole)]). fof(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(t6_boole,theorem,( ! [A] : ( v1_xboole_0(A) => A = k1_xboole_0 ) ), file(boole,t6_boole), [interesting(0.9),axiom,file(boole,t6_boole)]). fof(t7_boole,theorem,( ! [A,B] : ~ ( r2_hidden(A,B) & v1_xboole_0(B) ) ), file(boole,t7_boole), [interesting(0.9),axiom,file(boole,t7_boole)]). fof(t10_subset_1,theorem,( ! [A,B] : ( m1_subset_1(B,k1_zfmisc_1(A)) => ~ ( B != k1_xboole_0 & ! [C] : ( m1_subset_1(C,A) => ~ r2_hidden(C,B) ) ) ) ), file(subset_1,t10_subset_1), [interesting(0.9),axiom,file(subset_1,t10_subset_1)]). fof(e3_2__boolmark,plain,( ? [A] : ( m1_subset_1(A,c1_2__boolmark) & r2_hidden(A,k5_subset_1(c1_2__boolmark,c3_2__boolmark,c4_2__boolmark)) ) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark])],[commutativity_k3_xboole_0,idempotence_k3_xboole_0,reflexivity_r1_tarski,dt_k3_xboole_0,cc1_finseq_1,rc1_finseq_1,rc1_margrel1,rc1_subset_1,rc2_subset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_boole,t2_subset,t5_subset,t8_boole,commutativity_k5_subset_1,idempotence_k5_subset_1,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k5_subset_1,dt_m1_subset_1,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,fc1_margrel1,fc1_subset_1,fc2_finseq_1,t1_subset,t3_subset,t4_subset,t6_boole,t7_boole,e2_2__boolmark,t10_subset_1]), [interesting(0.8),file(boolmark,e3_2__boolmark),[file(boolmark,e3_2__boolmark)]]). fof(dt_c6_2__boolmark,plain,( m1_subset_1(c6_2__boolmark,c1_2__boolmark) ), inference(consider,[status(thm),assumptions([dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark])],[dh_c6_2__boolmark,e3_2__boolmark]), [interesting(0.8),file(boolmark,c6_2__boolmark),[file(boolmark,c6_2__boolmark)]]). fof(e4_2__boolmark,plain,( r2_hidden(c6_2__boolmark,k5_subset_1(c1_2__boolmark,c3_2__boolmark,c4_2__boolmark)) ), inference(consider,[status(thm),assumptions([dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark])],[dh_c6_2__boolmark,e3_2__boolmark]), [interesting(0.8),file(boolmark,e4_2__boolmark),[file(boolmark,e4_2__boolmark)]]). fof(d3_xboole_0,definition,( ! [A,B,C] : ( C = k3_xboole_0(A,B) <=> ! [D] : ( r2_hidden(D,C) <=> ( r2_hidden(D,A) & r2_hidden(D,B) ) ) ) ), file(xboole_0,d3_xboole_0), [interesting(0.9),axiom,file(xboole_0,d3_xboole_0)]). fof(e5_2__boolmark,plain, ( r2_hidden(c6_2__boolmark,c3_2__boolmark) & r2_hidden(c6_2__boolmark,c4_2__boolmark) ), inference(mizar_by,[status(thm),assumptions([dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark])],[cc1_finseq_1,rc1_finseq_1,rc1_margrel1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,reflexivity_r1_tarski,dt_k1_xboole_0,fc1_margrel1,fc2_finseq_1,t2_boole,existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_subset_1,fc1_subset_1,rc1_subset_1,rc2_subset_1,t2_subset,t3_subset,t4_subset,t5_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,antisymmetry_r2_hidden,redefinition_k5_subset_1,dt_k3_xboole_0,dt_k5_subset_1,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,dt_c6_2__boolmark,t1_subset,t7_boole,e4_2__boolmark,d3_xboole_0]), [interesting(0.8),file(boolmark,e5_2__boolmark),[file(boolmark,e5_2__boolmark)]]). fof(t43_funct_2,theorem,( ! [A,B,C,D] : ( ( v1_funct_1(D) & v1_funct_2(D,A,B) & m2_relset_1(D,A,B) ) => ( B != k1_xboole_0 => ! [E] : ( ? [F] : ( r2_hidden(F,A) & r2_hidden(F,C) & E = k1_funct_1(D,F) ) => r2_hidden(E,k9_relat_1(D,C)) ) ) ) ), file(funct_2,t43_funct_2), [interesting(0.9),axiom,file(funct_2,t43_funct_2)]). fof(e6_2__boolmark,plain, ( r2_hidden(k8_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c6_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c3_2__boolmark)) & r2_hidden(k8_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c6_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c4_2__boolmark)) ), inference(mizar_by,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark])],[reflexivity_r1_tarski,existence_m1_relset_1,existence_m1_subset_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,cc1_finseq_1,cc1_relset_1,fc1_subset_1,fc4_subset_1,rc1_finseq_1,rc1_margrel1,rc1_subset_1,rc2_subset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t4_subset,t5_subset,t8_boole,antisymmetry_r2_hidden,existence_m2_relset_1,redefinition_k2_funct_2,redefinition_k8_funct_2,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_xboole_0,dt_k2_funct_2,dt_k8_funct_2,dt_k9_relat_1,dt_m2_relset_1,dt_c1_2__boolmark,dt_c2_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,dt_c5_2__boolmark,dt_c6_2__boolmark,fc1_margrel1,fc2_finseq_1,t1_subset,t6_boole,t7_boole,e5_2__boolmark,t43_funct_2]), [interesting(0.8),file(boolmark,e6_2__boolmark),[file(boolmark,e6_2__boolmark)]]). fof(e7_2__boolmark,plain,( ~ $true ), inference(mizar_by,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark,e1_2__boolmark])],[reflexivity_r1_tarski,dt_k2_zfmisc_1,cc1_relset_1,fc4_subset_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_funct_1,dt_k1_zfmisc_1,dt_k9_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc1_finseq_1,fc1_subset_1,rc1_finseq_1,rc1_margrel1,rc1_subset_1,rc2_subset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,t2_subset,t3_subset,t4_subset,t5_subset,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,antisymmetry_r2_hidden,redefinition_k2_funct_2,redefinition_k5_subset_1,redefinition_k8_funct_2,dt_k1_xboole_0,dt_k2_funct_2,dt_k3_xboole_0,dt_k5_subset_1,dt_k8_funct_2,dt_c1_2__boolmark,dt_c2_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,dt_c5_2__boolmark,dt_c6_2__boolmark,fc1_margrel1,fc2_finseq_1,t1_subset,t2_boole,t6_boole,t7_boole,e6_2__boolmark,e1_2__boolmark,d3_xboole_0]), [interesting(0.8),file(boolmark,e7_2__boolmark),[file(boolmark,e7_2__boolmark)]]). fof(i6_2__boolmark,theorem,( $true ), introduced(tautology,[file(boolmark,i6_2__boolmark)]), [interesting(0.8),trivial,file(boolmark,i6_2__boolmark)]). fof(i5_2__boolmark,plain,( ~ $true ), inference(conclusion,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e2_2__boolmark,e1_2__boolmark])],[e7_2__boolmark,i6_2__boolmark]), [interesting(0.8),file(boolmark,i5_2__boolmark),[file(boolmark,i5_2__boolmark)]]). fof(i5_2_tmp__boolmark,plain, ( k5_subset_1(c1_2__boolmark,c3_2__boolmark,c4_2__boolmark) != k1_xboole_0 => ~ $true ), inference(discharge_asm,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e1_2__boolmark]),discharge_asm(discharge,[e2_2__boolmark])],[e2_2__boolmark,i5_2__boolmark]), [interesting(0.8),i4_2__boolmark]). fof(i4_2__boolmark,plain,( r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,e1_2__boolmark])],[i5_2_tmp__boolmark,d7_xboole_0,dt_k1_zfmisc_1,dt_m1_subset_1,cc1_finseq_1,fc1_subset_1,rc1_finseq_1,rc1_margrel1,rc1_subset_1,rc2_subset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,symmetry_r1_xboole_0,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k3_xboole_0,dt_k5_subset_1,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,fc1_margrel1,fc2_finseq_1]), [interesting(0.8),file(boolmark,i4_2__boolmark),[file(boolmark,i4_2__boolmark)]]). fof(i4_2_tmp__boolmark,plain, ( k5_subset_1(c2_2__boolmark,k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c4_2__boolmark)) = k1_xboole_0 => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark]),discharge_asm(discharge,[e1_2__boolmark])],[e1_2__boolmark,i4_2__boolmark]), [interesting(0.8),i3_2__boolmark]). fof(i3_2__boolmark,plain, ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ), inference(mizar_def_expansion,[status(thm),assumptions([dt_c2_2__boolmark,dt_c5_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark])],[i4_2_tmp__boolmark,d7_xboole_0,dt_k2_zfmisc_1,cc1_relset_1,fc4_subset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k9_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc1_finseq_1,fc1_subset_1,rc1_finseq_1,rc1_margrel1,rc1_subset_1,rc2_subset_1,rc3_finseq_1,rc6_finseq_1,rc7_finseq_1,rc8_finseq_1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,commutativity_k5_subset_1,idempotence_k5_subset_1,symmetry_r1_xboole_0,redefinition_k2_funct_2,redefinition_k5_subset_1,dt_k1_xboole_0,dt_k2_funct_2,dt_k3_xboole_0,dt_k5_subset_1,dt_c1_2__boolmark,dt_c2_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark,dt_c5_2__boolmark,fc1_margrel1,fc2_finseq_1]), [interesting(0.8),file(boolmark,i3_2__boolmark),[file(boolmark,i3_2__boolmark)]]). fof(i3_2_tmp__boolmark,plain, ( ( v1_funct_1(c5_2__boolmark) & v1_funct_2(c5_2__boolmark,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(c5_2__boolmark,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,c5_2__boolmark,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark]),discharge_asm(discharge,[dt_c5_2__boolmark])],[dt_c5_2__boolmark,i3_2__boolmark]), [interesting(0.8),i2_2__boolmark]). fof(i2_2__boolmark,plain,( ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(A,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ) ) ), inference(let,[status(thm),assumptions([dt_c2_2__boolmark,dt_c1_2__boolmark,dt_c3_2__boolmark,dt_c4_2__boolmark])],[i3_2_tmp__boolmark,dh_c5_2__boolmark]), [interesting(0.8),file(boolmark,i2_2__boolmark),[file(boolmark,i2_2__boolmark)]]). fof(i2_2_tmp__boolmark,plain, ( ( m1_subset_1(c3_2__boolmark,k1_zfmisc_1(c1_2__boolmark)) & m1_subset_1(c4_2__boolmark,k1_zfmisc_1(c1_2__boolmark)) ) => ! [A] : ( ( v1_funct_1(A) & v1_funct_2(A,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(A,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c3_2__boolmark),k2_funct_2(c1_2__boolmark,c2_2__boolmark,A,c4_2__boolmark)) => r1_xboole_0(c3_2__boolmark,c4_2__boolmark) ) ) ), inference(discharge_asm,[status(thm),assumptions([dt_c2_2__boolmark,dt_c1_2__boolmark]),discharge_asm(discharge,[dt_c3_2__boolmark,dt_c4_2__boolmark])],[dt_c3_2__boolmark,dt_c4_2__boolmark,i2_2__boolmark]), [interesting(0.8),i1_2__boolmark]). fof(i1_2__boolmark,plain,( ! [A] : ( m1_subset_1(A,k1_zfmisc_1(c1_2__boolmark)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(c1_2__boolmark)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(C,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,A),k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,B)) => r1_xboole_0(A,B) ) ) ) ) ), inference(let,[status(thm),assumptions([dt_c2_2__boolmark,dt_c1_2__boolmark])],[i2_2_tmp__boolmark,dh_c3_2__boolmark,dh_c4_2__boolmark]), [interesting(0.8),file(boolmark,i1_2__boolmark),[file(boolmark,i1_2__boolmark)]]). fof(i1_2_tmp__boolmark,plain, ( ( ~ v1_xboole_0(c1_2__boolmark) & ~ v1_xboole_0(c2_2__boolmark) ) => ! [A] : ( m1_subset_1(A,k1_zfmisc_1(c1_2__boolmark)) => ! [B] : ( m1_subset_1(B,k1_zfmisc_1(c1_2__boolmark)) => ! [C] : ( ( v1_funct_1(C) & v1_funct_2(C,c1_2__boolmark,c2_2__boolmark) & m2_relset_1(C,c1_2__boolmark,c2_2__boolmark) ) => ( r1_xboole_0(k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,A),k2_funct_2(c1_2__boolmark,c2_2__boolmark,C,B)) => r1_xboole_0(A,B) ) ) ) ) ), inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_2__boolmark,dt_c2_2__boolmark])],[dt_c1_2__boolmark,dt_c2_2__boolmark,i1_2__boolmark]), [interesting(1),t2_boolmark]). fof(t2_boolmark,theorem,( ! [A] : ( ~ v1_xboole_0(A) => ! [B] : ( ~ v1_xboole_0(B) => ! [C] : ( m1_subset_1(C,k1_zfmisc_1(A)) => ! [D] : ( m1_subset_1(D,k1_zfmisc_1(A)) => ! [E] : ( ( v1_funct_1(E) & v1_funct_2(E,A,B) & m2_relset_1(E,A,B) ) => ( r1_xboole_0(k2_funct_2(A,B,E,C),k2_funct_2(A,B,E,D)) => r1_xboole_0(C,D) ) ) ) ) ) ) ), inference(let,[status(thm),assumptions([])],[i1_2_tmp__boolmark,dh_c1_2__boolmark,dh_c2_2__boolmark]), [interesting(1),file(boolmark,t2_boolmark),[file(boolmark,t2_boolmark)]]).