% Mizar ND problem: t7_algspec1,algspec1,202,47 
fof(dh_c1_8__algspec1,definition,
    ( ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A) )
       => k2_relat_1(k1_algspec1(c1_8__algspec1,A)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(A)),k9_relat_1(A,c1_8__algspec1)) )
   => ! [B,C] : 
        ( ( v1_relat_1(C)
          & v1_funct_1(C) )
       => k2_relat_1(k1_algspec1(B,C)) = k2_xboole_0(k4_xboole_0(B,k1_relat_1(C)),k9_relat_1(C,B)) ) ),
    introduced(definition,[new_symbol(c1_8__algspec1),file(algspec1,c1_8__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c1_8__algspec1)]).

fof(dh_c2_8__algspec1,definition,
    ( ( ( v1_relat_1(c2_8__algspec1)
        & v1_funct_1(c2_8__algspec1) )
     => k2_relat_1(k1_algspec1(c1_8__algspec1,c2_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(c2_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A) )
       => k2_relat_1(k1_algspec1(c1_8__algspec1,A)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(A)),k9_relat_1(A,c1_8__algspec1)) ) ),
    introduced(definition,[new_symbol(c2_8__algspec1),file(algspec1,c2_8__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c2_8__algspec1)]).

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(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_k6_relat_1,axiom,(
    ! [A] : v1_relat_1(k6_relat_1(A)) ),
    file(relat_1,k6_relat_1),
    [interesting(0.9),axiom,file(relat_1,k6_relat_1)]).

fof(dt_m1_pboole,axiom,(
    ! [A,B] : 
      ( m1_pboole(B,A)
     => ( v1_relat_1(B)
        & v1_funct_1(B) ) ) ),
    file(pboole,m1_pboole),
    [interesting(0.9),axiom,file(pboole,m1_pboole)]).

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(fc1_relat_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k3_xboole_0(A,B)) ) ),
    file(relat_1,fc1_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc1_relat_1)]).

fof(fc2_funct_1,theorem,(
    ! [A] : 
      ( v1_relat_1(k6_relat_1(A))
      & v1_funct_1(k6_relat_1(A)) ) ),
    file(funct_1,fc2_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc2_funct_1)]).

fof(fc2_relat_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k2_xboole_0(A,B)) ) ),
    file(relat_1,fc2_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc2_relat_1)]).

fof(fc3_relat_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k4_xboole_0(A,B)) ) ),
    file(relat_1,fc3_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc3_relat_1)]).

fof(fc4_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_relat_1(k7_relat_1(A,B))
        & v1_funct_1(k7_relat_1(A,B)) ) ) ),
    file(funct_1,fc4_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc4_funct_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(commutativity_k2_xboole_0,theorem,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(idempotence_k2_xboole_0,theorem,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(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(redefinition_k10_relset_1,definition,(
    ! [A,B,C,D] : 
      ( m1_relset_1(C,A,B)
     => k10_relset_1(A,B,C,D) = k9_relat_1(C,D) ) ),
    file(relset_1,k10_relset_1),
    [interesting(0.9),axiom,file(relset_1,k10_relset_1)]).

fof(redefinition_k6_partfun1,definition,(
    ! [A] : k6_partfun1(A) = k6_relat_1(A) ),
    file(partfun1,k6_partfun1),
    [interesting(0.9),axiom,file(partfun1,k6_partfun1)]).

fof(dt_k10_relset_1,axiom,(
    ! [A,B,C,D] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k10_relset_1(A,B,C,D),k1_zfmisc_1(B)) ) ),
    file(relset_1,k10_relset_1),
    [interesting(0.9),axiom,file(relset_1,k10_relset_1)]).

fof(dt_k1_algspec1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => m1_pboole(k1_algspec1(A,B),A) ) ),
    file(algspec1,k1_algspec1),
    [interesting(0.9),axiom,file(algspec1,k1_algspec1)]).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    [interesting(0.9),axiom,file(relat_1,k1_relat_1)]).

fof(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    [interesting(0.9),axiom,file(relat_1,k2_relat_1)]).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(dt_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k4_xboole_0)]).

fof(dt_k6_partfun1,axiom,(
    ! [A] : 
      ( v1_partfun1(k6_partfun1(A),A,A)
      & m2_relset_1(k6_partfun1(A),A,A) ) ),
    file(partfun1,k6_partfun1),
    [interesting(0.9),axiom,file(partfun1,k6_partfun1)]).

fof(dt_k7_relat_1,axiom,(
    ! [A,B] : 
      ( v1_relat_1(A)
     => v1_relat_1(k7_relat_1(A,B)) ) ),
    file(relat_1,k7_relat_1),
    [interesting(0.9),axiom,file(relat_1,k7_relat_1)]).

fof(dt_k9_relat_1,axiom,(
    $true ),
    file(relat_1,k9_relat_1),
    [interesting(0.9),axiom,file(relat_1,k9_relat_1)]).

fof(dt_c1_8__algspec1,assumption,(
    $true ),
    introduced(assumption,[file(algspec1,c1_8__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c1_8__algspec1)]).

fof(dt_c2_8__algspec1,assumption,
    ( v1_relat_1(c2_8__algspec1)
    & v1_funct_1(c2_8__algspec1) ),
    introduced(assumption,[file(algspec1,c2_8__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c2_8__algspec1)]).

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_pboole,axiom,(
    ! [A] : 
    ? [B] : m1_pboole(B,A) ),
    file(pboole,m1_pboole),
    [interesting(0.9),axiom,file(pboole,m1_pboole)]).

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(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(idempotence_k1_funct_4,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_funct_4(A,A) = A ) ),
    file(funct_4,k1_funct_4),
    [interesting(0.9),axiom,file(funct_4,k1_funct_4)]).

fof(redefinition_k4_relset_1,definition,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => k4_relset_1(A,B,C) = k1_relat_1(C) ) ),
    file(relset_1,k4_relset_1),
    [interesting(0.9),axiom,file(relset_1,k4_relset_1)]).

fof(dt_k1_funct_4,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k1_funct_4(A,B))
        & v1_funct_1(k1_funct_4(A,B)) ) ) ),
    file(funct_4,k1_funct_4),
    [interesting(0.9),axiom,file(funct_4,k1_funct_4)]).

fof(dt_k4_relset_1,axiom,(
    ! [A,B,C] : 
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) ),
    file(relset_1,k4_relset_1),
    [interesting(0.9),axiom,file(relset_1,k4_relset_1)]).

fof(d1_algspec1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k1_algspec1(A,B) = k1_funct_4(k6_partfun1(A),k7_relat_1(B,A)) ) ),
    file(algspec1,d1_algspec1),
    [interesting(0.9),axiom,file(algspec1,d1_algspec1)]).

fof(t71_relat_1,theorem,(
    ! [A] : 
      ( k1_relat_1(k6_relat_1(A)) = A
      & k2_relat_1(k6_relat_1(A)) = A ) ),
    file(relat_1,t71_relat_1),
    [interesting(0.9),axiom,file(relat_1,t71_relat_1)]).

fof(e2_8__algspec1,plain,
    ( k1_algspec1(c1_8__algspec1,c2_8__algspec1) = k1_funct_4(k6_partfun1(c1_8__algspec1),k7_relat_1(c2_8__algspec1,c1_8__algspec1))
    & k4_relset_1(c1_8__algspec1,c1_8__algspec1,k6_partfun1(c1_8__algspec1)) = c1_8__algspec1 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[reflexivity_r1_tarski,dt_k2_zfmisc_1,cc1_relset_1,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc4_funct_1,rc1_funct_1,t3_subset,idempotence_k1_funct_4,redefinition_k4_relset_1,redefinition_k6_partfun1,dt_k1_algspec1,dt_k1_funct_4,dt_k1_relat_1,dt_k2_relat_1,dt_k4_relset_1,dt_k6_partfun1,dt_k6_relat_1,dt_k7_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,fc2_funct_1,d1_algspec1,t71_relat_1]),
    [interesting(0.8),file(algspec1,e2_8__algspec1),[file(algspec1,e2_8__algspec1)]]).

fof(t12_frechet,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => k2_relat_1(k1_funct_4(A,B)) = k2_xboole_0(k9_relat_1(A,k4_xboole_0(k1_relat_1(A),k1_relat_1(B))),k2_relat_1(B)) ) ) ),
    file(frechet,t12_frechet),
    [interesting(0.9),axiom,file(frechet,t12_frechet)]).

fof(e1_8_1__algspec1,plain,(
    k2_relat_1(k1_algspec1(c1_8__algspec1,c2_8__algspec1)) = k2_xboole_0(k10_relset_1(c1_8__algspec1,c1_8__algspec1,k6_partfun1(c1_8__algspec1),k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1)))),k2_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[reflexivity_r1_tarski,dt_k2_zfmisc_1,cc1_relset_1,existence_m1_pboole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k6_relat_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc2_funct_1,t3_subset,idempotence_k1_funct_4,commutativity_k2_xboole_0,idempotence_k2_xboole_0,redefinition_k10_relset_1,redefinition_k4_relset_1,redefinition_k6_partfun1,dt_k10_relset_1,dt_k1_algspec1,dt_k1_funct_4,dt_k1_relat_1,dt_k2_relat_1,dt_k2_xboole_0,dt_k4_relset_1,dt_k4_xboole_0,dt_k6_partfun1,dt_k7_relat_1,dt_k9_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,fc2_relat_1,fc3_relat_1,fc4_funct_1,rc1_funct_1,d1_algspec1,e2_8__algspec1,t12_frechet]),
    [interesting(0.65),file(algspec1,e1_8_1__algspec1),[file(algspec1,e1_8_1__algspec1)]]).

fof(t148_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k2_relat_1(k7_relat_1(B,A)) = k9_relat_1(B,A) ) ),
    file(relat_1,t148_relat_1),
    [interesting(0.9),axiom,file(relat_1,t148_relat_1)]).

fof(e2_8_1__algspec1,plain,(
    k2_xboole_0(k10_relset_1(c1_8__algspec1,c1_8__algspec1,k6_partfun1(c1_8__algspec1),k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1)))),k2_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1))) = k2_xboole_0(k10_relset_1(c1_8__algspec1,c1_8__algspec1,k6_partfun1(c1_8__algspec1),k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1)))),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[reflexivity_r1_tarski,dt_k2_zfmisc_1,cc1_relset_1,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k6_relat_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc2_funct_1,fc4_funct_1,rc1_funct_1,t3_subset,commutativity_k2_xboole_0,idempotence_k2_xboole_0,redefinition_k10_relset_1,redefinition_k6_partfun1,dt_k10_relset_1,dt_k1_relat_1,dt_k2_relat_1,dt_k2_xboole_0,dt_k4_xboole_0,dt_k6_partfun1,dt_k7_relat_1,dt_k9_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,fc2_relat_1,fc3_relat_1,t148_relat_1]),
    [interesting(0.65),file(algspec1,e2_8_1__algspec1),[file(algspec1,e2_8_1__algspec1)]]).

fof(t36_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    file(xboole_1,t36_xboole_1),
    [interesting(0.9),axiom,file(xboole_1,t36_xboole_1)]).

fof(e1_8__algspec1,plain,(
    r1_tarski(k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1))),c1_8__algspec1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[existence_m1_subset_1,dt_k1_zfmisc_1,dt_m1_subset_1,fc3_relat_1,fc4_funct_1,rc1_funct_1,reflexivity_r1_tarski,dt_k1_relat_1,dt_k4_xboole_0,dt_k7_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,t3_subset,t36_xboole_1]),
    [interesting(0.8),file(algspec1,e1_8__algspec1),[file(algspec1,e1_8__algspec1)]]).

fof(t162_funct_1,theorem,(
    ! [A,B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k9_relat_1(k6_relat_1(A),B) = B ) ),
    file(funct_1,t162_funct_1),
    [interesting(0.9),axiom,file(funct_1,t162_funct_1)]).

fof(e3_8_1__algspec1,plain,(
    k2_xboole_0(k10_relset_1(c1_8__algspec1,c1_8__algspec1,k6_partfun1(c1_8__algspec1),k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1)))),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1))),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[dt_k2_zfmisc_1,cc1_relset_1,existence_m1_relset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_m1_relset_1,dt_m2_relset_1,fc2_relat_1,fc3_relat_1,fc4_funct_1,rc1_funct_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,existence_m1_subset_1,redefinition_k10_relset_1,redefinition_k6_partfun1,dt_k10_relset_1,dt_k1_relat_1,dt_k1_zfmisc_1,dt_k2_xboole_0,dt_k4_xboole_0,dt_k6_partfun1,dt_k6_relat_1,dt_k7_relat_1,dt_k9_relat_1,dt_m1_subset_1,dt_c1_8__algspec1,dt_c2_8__algspec1,fc2_funct_1,t3_subset,e1_8__algspec1,t162_funct_1]),
    [interesting(0.65),file(algspec1,e3_8_1__algspec1),[file(algspec1,e3_8_1__algspec1)]]).

fof(t90_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ),
    file(relat_1,t90_relat_1),
    [interesting(0.9),axiom,file(relat_1,t90_relat_1)]).

fof(e4_8_1__algspec1,plain,(
    k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(k7_relat_1(c2_8__algspec1,c1_8__algspec1))),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k3_xboole_0(k1_relat_1(c2_8__algspec1),c1_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[fc4_funct_1,rc1_funct_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,dt_k1_relat_1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_xboole_0,dt_k7_relat_1,dt_k9_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,fc1_relat_1,fc2_relat_1,fc3_relat_1,t90_relat_1]),
    [interesting(0.65),file(algspec1,e4_8_1__algspec1),[file(algspec1,e4_8_1__algspec1)]]).

fof(t47_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(A,k3_xboole_0(A,B)) = k4_xboole_0(A,B) ),
    file(xboole_1,t47_xboole_1),
    [interesting(0.9),axiom,file(xboole_1,t47_xboole_1)]).

fof(e5_8_1__algspec1,plain,(
    k2_xboole_0(k4_xboole_0(c1_8__algspec1,k3_xboole_0(k1_relat_1(c2_8__algspec1),c1_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(c2_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[fc1_relat_1,fc2_relat_1,fc3_relat_1,rc1_funct_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,dt_k1_relat_1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_xboole_0,dt_k9_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,t47_xboole_1]),
    [interesting(0.65),file(algspec1,e5_8_1__algspec1),[file(algspec1,e5_8_1__algspec1)]]).

fof(e3_8__algspec1,plain,(
    k2_relat_1(k1_algspec1(c1_8__algspec1,c2_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(c2_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[dt_k2_zfmisc_1,cc1_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_k6_relat_1,dt_m1_pboole,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_relat_1,fc2_funct_1,fc2_relat_1,fc3_relat_1,fc4_funct_1,rc1_funct_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,commutativity_k3_xboole_0,idempotence_k3_xboole_0,redefinition_k10_relset_1,redefinition_k6_partfun1,dt_k10_relset_1,dt_k1_algspec1,dt_k1_relat_1,dt_k2_relat_1,dt_k2_xboole_0,dt_k3_xboole_0,dt_k4_xboole_0,dt_k6_partfun1,dt_k7_relat_1,dt_k9_relat_1,dt_c1_8__algspec1,dt_c2_8__algspec1,e1_8_1__algspec1,e2_8_1__algspec1,e3_8_1__algspec1,e4_8_1__algspec1,e5_8_1__algspec1]),
    [interesting(0.8),file(algspec1,e3_8__algspec1),[file(algspec1,e3_8__algspec1)]]).

fof(i3_8__algspec1,theorem,(
    $true ),
    introduced(tautology,[file(algspec1,i3_8__algspec1)]),
    [interesting(0.8),trivial,file(algspec1,i3_8__algspec1)]).

fof(i2_8__algspec1,plain,(
    k2_relat_1(k1_algspec1(c1_8__algspec1,c2_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(c2_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_8__algspec1,dt_c2_8__algspec1])],[e3_8__algspec1,i3_8__algspec1]),
    [interesting(0.8),file(algspec1,i2_8__algspec1),[file(algspec1,i2_8__algspec1)]]).

fof(i2_8_tmp__algspec1,plain,
    ( ( v1_relat_1(c2_8__algspec1)
      & v1_funct_1(c2_8__algspec1) )
   => k2_relat_1(k1_algspec1(c1_8__algspec1,c2_8__algspec1)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(c2_8__algspec1)),k9_relat_1(c2_8__algspec1,c1_8__algspec1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_8__algspec1]),discharge_asm(discharge,[dt_c2_8__algspec1])],[dt_c2_8__algspec1,i2_8__algspec1]),
    [interesting(0.8),i1_8__algspec1]).

fof(i1_8__algspec1,plain,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k2_relat_1(k1_algspec1(c1_8__algspec1,A)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(A)),k9_relat_1(A,c1_8__algspec1)) ) ),
    inference(let,[status(thm),assumptions([dt_c1_8__algspec1])],[i2_8_tmp__algspec1,dh_c2_8__algspec1]),
    [interesting(0.8),file(algspec1,i1_8__algspec1),[file(algspec1,i1_8__algspec1)]]).

fof(i1_8_tmp__algspec1,plain,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => k2_relat_1(k1_algspec1(c1_8__algspec1,A)) = k2_xboole_0(k4_xboole_0(c1_8__algspec1,k1_relat_1(A)),k9_relat_1(A,c1_8__algspec1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_8__algspec1])],[dt_c1_8__algspec1,i1_8__algspec1]),
    [interesting(1),t7_algspec1]).

fof(t7_algspec1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => k2_relat_1(k1_algspec1(A,B)) = k2_xboole_0(k4_xboole_0(A,k1_relat_1(B)),k9_relat_1(B,A)) ) ),
    inference(let,[status(thm),assumptions([])],[i1_8_tmp__algspec1,dh_c1_8__algspec1]),
    [interesting(1),file(algspec1,t7_algspec1),[file(algspec1,t7_algspec1)]]).