% Mizar ND problem: t3_funct_2,funct_2,82,56 
fof(dh_c1_5__funct_2,definition,
    ( ( ( v1_relat_1(c1_5__funct_2)
        & v1_funct_1(c1_5__funct_2) )
     => ( v1_funct_1(c1_5__funct_2)
        & v1_funct_2(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2))
        & m2_relset_1(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A) )
       => ( v1_funct_1(A)
          & v1_funct_2(A,k1_relat_1(A),k2_relat_1(A))
          & m2_relset_1(A,k1_relat_1(A),k2_relat_1(A)) ) ) ),
    introduced(definition,[new_symbol(c1_5__funct_2),file(funct_2,c1_5__funct_2)]),
    [interesting(0.8),axiom,file(funct_2,c1_5__funct_2)]).

fof(rc3_relat_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A) ) ),
    file(relat_1,rc3_relat_1),
    [interesting(0.9),axiom,file(relat_1,rc3_relat_1)]).

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(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(fc12_relat_1,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0) ),
    file(relat_1,fc12_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc12_relat_1)]).

fof(fc1_xboole_0,theorem,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]).

fof(fc4_relat_1,theorem,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0) ),
    file(relat_1,fc4_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc4_relat_1)]).

fof(rc1_partfun1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A)
      & v1_xboole_0(A) ) ),
    file(partfun1,rc1_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc1_partfun1)]).

fof(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(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(t4_subset,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    [interesting(0.9),axiom,file(subset,t4_subset)]).

fof(t5_subset,theorem,(
    ! [A,B,C] : 
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    [interesting(0.9),axiom,file(subset,t5_subset)]).

fof(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

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_relat_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ),
    file(relat_1,cc1_relat_1),
    [interesting(0.9),axiom,file(relat_1,cc1_relat_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(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(fc5_relat_1,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A) )
     => ~ v1_xboole_0(k1_relat_1(A)) ) ),
    file(relat_1,fc5_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc5_relat_1)]).

fof(fc6_relat_1,theorem,(
    ! [A] : 
      ( ( ~ v1_xboole_0(A)
        & v1_relat_1(A) )
     => ~ v1_xboole_0(k2_relat_1(A)) ) ),
    file(relat_1,fc6_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc6_relat_1)]).

fof(fc7_relat_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_xboole_0(k1_relat_1(A))
        & v1_relat_1(k1_relat_1(A)) ) ) ),
    file(relat_1,fc7_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc7_relat_1)]).

fof(fc8_relat_1,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => ( v1_xboole_0(k2_relat_1(A))
        & v1_relat_1(k2_relat_1(A)) ) ) ),
    file(relat_1,fc8_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc8_relat_1)]).

fof(rc1_relat_1,theorem,(
    ? [A] : 
      ( v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc1_relat_1),
    [interesting(0.9),axiom,file(relat_1,rc1_relat_1)]).

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(rc1_xboole_0,theorem,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]).

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_relat_1,theorem,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A) ) ),
    file(relat_1,rc2_relat_1),
    [interesting(0.9),axiom,file(relat_1,rc2_relat_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(rc2_xboole_0,theorem,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]).

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(t6_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

fof(t7_boole,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    [interesting(0.9),axiom,file(boole,t7_boole)]).

fof(t8_boole,theorem,(
    ! [A,B] : 
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    [interesting(0.9),axiom,file(boole,t8_boole)]).

fof(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(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_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_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(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(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_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_funct_2,theorem,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C)
      & v1_funct_2(C,A,B) ) ),
    file(funct_2,rc1_funct_2),
    [interesting(0.9),axiom,file(funct_2,rc1_funct_2)]).

fof(rc2_partfun1,theorem,(
    ! [A,B] : 
    ? [C] : 
      ( m1_relset_1(C,A,B)
      & v1_relat_1(C)
      & v1_funct_1(C) ) ),
    file(partfun1,rc2_partfun1),
    [interesting(0.9),axiom,file(partfun1,rc2_partfun1)]).

fof(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(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(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_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_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_c1_5__funct_2,assumption,
    ( v1_relat_1(c1_5__funct_2)
    & v1_funct_1(c1_5__funct_2) ),
    introduced(assumption,[file(funct_2,c1_5__funct_2)]),
    [interesting(0.8),axiom,file(funct_2,c1_5__funct_2)]).

fof(de_c2_5__funct_2,definition,(
    c2_5__funct_2 = c1_5__funct_2 ),
    introduced(definition,[new_symbol(c2_5__funct_2),file(funct_2,c2_5__funct_2)]),
    [interesting(0.8),axiom,file(funct_2,c2_5__funct_2)]).

fof(t9_relset_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(k1_relat_1(B),A)
       => m2_relset_1(B,A,k2_relat_1(B)) ) ) ),
    file(relset_1,t9_relset_1),
    [interesting(0.9),axiom,file(relset_1,t9_relset_1)]).

fof(e1_5__funct_2,plain,(
    m2_relset_1(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__funct_2])],[rc3_relat_1,rc4_funct_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_partfun1,rc3_funct_1,t1_subset,t4_subset,t5_subset,t8_boole,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc4_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,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_relset_1,fc1_subset_1,rc1_funct_1,rc2_partfun1,reflexivity_r1_tarski,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k2_relat_1,dt_m2_relset_1,dt_c1_5__funct_2,t3_subset,t9_relset_1]),
    [interesting(0.8),file(funct_2,e1_5__funct_2),[file(funct_2,e1_5__funct_2)]]).

fof(dt_c2_5__funct_2,plain,(
    m2_relset_1(c2_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__funct_2])],[rc3_relat_1,rc4_funct_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_partfun1,rc3_funct_1,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc4_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,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_relset_1,fc1_subset_1,rc1_funct_1,rc2_partfun1,t3_subset,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k2_relat_1,dt_m2_relset_1,dt_c1_5__funct_2,de_c2_5__funct_2,e1_5__funct_2]),
    [interesting(0.8),file(funct_2,c2_5__funct_2),[file(funct_2,c2_5__funct_2)]]).

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_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_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.9),axiom,file(funct_2,d1_funct_2)]).

fof(e1_5_1_1_1__funct_2,assumption,
    ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
   => k1_relat_1(c1_5__funct_2) = k1_xboole_0 ),
    introduced(assumption,[file(funct_2,e1_5_1_1_1__funct_2)]),
    [interesting(0.35),axiom,file(funct_2,e1_5_1_1_1__funct_2)]).

fof(e2_5_1_1_1__funct_2,plain,(
    k1_relat_1(c1_5__funct_2) = k4_relset_1(k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2),c2_5__funct_2) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__funct_2])],[rc3_relat_1,rc4_funct_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_partfun1,rc3_funct_1,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,dt_k2_zfmisc_1,cc1_funct_1,cc1_relat_1,cc1_relset_1,cc2_funct_1,fc4_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,existence_m1_relset_1,existence_m1_subset_1,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,rc1_funct_1,rc2_partfun1,t3_subset,redefinition_k4_relset_1,dt_k1_relat_1,dt_k2_relat_1,dt_k4_relset_1,dt_c1_5__funct_2,dt_c2_5__funct_2,de_c2_5__funct_2]),
    [interesting(0.35),file(funct_2,e2_5_1_1_1__funct_2),[file(funct_2,e2_5_1_1_1__funct_2)]]).

fof(i2_5_1_1_1__funct_2,theorem,(
    $true ),
    introduced(tautology,[file(funct_2,i2_5_1_1_1__funct_2)]),
    [interesting(0.35),trivial,file(funct_2,i2_5_1_1_1__funct_2)]).

fof(i1_5_1_1_1__funct_2,plain,(
    k1_relat_1(c1_5__funct_2) = k4_relset_1(k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2),c2_5__funct_2) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__funct_2])],[rc1_partfun1,rc3_funct_1,dt_k2_zfmisc_1,cc1_funct_1,cc1_relat_1,cc1_relset_1,cc2_funct_1,fc4_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,fc1_subset_1,rc1_funct_1,rc2_partfun1,redefinition_k4_relset_1,dt_k1_relat_1,dt_k2_relat_1,dt_k4_relset_1,dt_c1_5__funct_2,dt_c2_5__funct_2,e2_5_1_1_1__funct_2,i2_5_1_1_1__funct_2]),
    [interesting(0.35),file(funct_2,i1_5_1_1_1__funct_2),[file(funct_2,i1_5_1_1_1__funct_2)]]).

fof(i1_5_1_1_1_tmp__funct_2,plain,
    ( ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
     => k1_relat_1(c1_5__funct_2) = k1_xboole_0 )
   => k1_relat_1(c1_5__funct_2) = k4_relset_1(k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2),c2_5__funct_2) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__funct_2]),discharge_asm(discharge,[e1_5_1_1_1__funct_2])],[e1_5_1_1_1__funct_2,i1_5_1_1_1__funct_2]),
    [interesting(0.5),i1_5_1_1__funct_2]).

fof(i1_5_1_1__funct_2,plain,
    ( ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
     => k1_relat_1(c1_5__funct_2) = k1_xboole_0 )
   => k1_relat_1(c1_5__funct_2) = k4_relset_1(k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2),c2_5__funct_2) ),
    inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_5__funct_2])],[i1_5_1_1_1_tmp__funct_2,dt_k2_zfmisc_1,cc1_relset_1,fc4_subset_1,rc1_partfun1,rc3_funct_1,redefinition_m2_relset_1,dt_k1_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,dt_m2_relset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc1_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_funct_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_partfun1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,rc3_relat_1,rc4_funct_1,redefinition_k4_relset_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k2_relat_1,dt_k4_relset_1,dt_c1_5__funct_2,dt_c2_5__funct_2,fc12_relat_1,fc1_xboole_0,fc4_relat_1]),
    [interesting(0.5),file(funct_2,i1_5_1_1__funct_2),[file(funct_2,i1_5_1_1__funct_2)]]).

fof(e1_5_1_1_2__funct_2,assumption,
    ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
    & k1_relat_1(c1_5__funct_2) != k1_xboole_0 ),
    introduced(assumption,[file(funct_2,e1_5_1_1_2__funct_2)]),
    [interesting(0.35),axiom,file(funct_2,e1_5_1_1_2__funct_2)]).

fof(t65_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( k1_relat_1(A) = k1_xboole_0
      <=> k2_relat_1(A) = k1_xboole_0 ) ) ),
    file(relat_1,t65_relat_1),
    [interesting(0.9),axiom,file(relat_1,t65_relat_1)]).

fof(e2_5_1_1_2__funct_2,plain,(
    c2_5__funct_2 = k1_xboole_0 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__funct_2,e1_5_1_1_2__funct_2])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,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_relset_1,fc1_subset_1,fc4_subset_1,rc1_partfun1,rc1_subset_1,rc2_partfun1,rc2_subset_1,rc3_funct_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,existence_m2_relset_1,redefinition_m2_relset_1,dt_m2_relset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_funct_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,rc3_relat_1,rc4_funct_1,t7_boole,t8_boole,dt_k1_relat_1,dt_k1_xboole_0,dt_k2_relat_1,dt_c1_5__funct_2,dt_c2_5__funct_2,de_c2_5__funct_2,fc12_relat_1,fc1_xboole_0,fc4_relat_1,t6_boole,e1_5_1_1_2__funct_2,t65_relat_1]),
    [interesting(0.35),file(funct_2,e2_5_1_1_2__funct_2),[file(funct_2,e2_5_1_1_2__funct_2)]]).

fof(i2_5_1_1_2__funct_2,theorem,(
    $true ),
    introduced(tautology,[file(funct_2,i2_5_1_1_2__funct_2)]),
    [interesting(0.35),trivial,file(funct_2,i2_5_1_1_2__funct_2)]).

fof(i1_5_1_1_2__funct_2,plain,(
    c2_5__funct_2 = k1_xboole_0 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__funct_2,e1_5_1_1_2__funct_2])],[e2_5_1_1_2__funct_2,i2_5_1_1_2__funct_2]),
    [interesting(0.35),file(funct_2,i1_5_1_1_2__funct_2),[file(funct_2,i1_5_1_1_2__funct_2)]]).

fof(i2_5_1_1__funct_2,plain,
    ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
   => ( k1_relat_1(c1_5__funct_2) = k1_xboole_0
      | c2_5__funct_2 = k1_xboole_0 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__funct_2]),discharge_asm(discharge,[e1_5_1_1_2__funct_2])],[e1_5_1_1_2__funct_2,i1_5_1_1_2__funct_2]),
    [interesting(0.5),file(funct_2,i2_5_1_1__funct_2),[file(funct_2,i2_5_1_1__funct_2)]]).

fof(e1_5_1_1__funct_2,plain,(
    ~ ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
      & k1_relat_1(c1_5__funct_2) != k1_xboole_0
      & ~ ( k2_relat_1(c1_5__funct_2) = k1_xboole_0
          & k1_relat_1(c1_5__funct_2) != k1_xboole_0 ) ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__funct_2])],[existence_m1_subset_1,dt_m1_subset_1,t2_subset,antisymmetry_r2_hidden,rc1_partfun1,rc3_funct_1,t1_subset,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_funct_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,rc3_relat_1,rc4_funct_1,t7_boole,t8_boole,dt_k1_relat_1,dt_k1_xboole_0,dt_k2_relat_1,dt_c1_5__funct_2,fc12_relat_1,fc1_xboole_0,fc4_relat_1,t6_boole]),
    [interesting(0.5),file(funct_2,e1_5_1_1__funct_2),[file(funct_2,e1_5_1_1__funct_2)]]).

fof(e2_5__funct_2,plain,(
    v1_funct_2(c2_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ),
    inference(percases,[status(thm),assumptions([dt_c1_5__funct_2])],[rc1_partfun1,rc3_funct_1,dt_k1_zfmisc_1,dt_k2_zfmisc_1,dt_m1_relset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc1_relset_1,cc2_funct_1,fc1_subset_1,fc4_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_funct_1,rc1_funct_2,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_partfun1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,rc3_relat_1,rc4_funct_1,redefinition_k4_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k1_xboole_0,dt_k2_relat_1,dt_k4_relset_1,dt_m2_relset_1,dt_c1_5__funct_2,dt_c2_5__funct_2,fc12_relat_1,fc1_xboole_0,fc4_relat_1,d1_funct_2,i1_5_1_1__funct_2,i2_5_1_1__funct_2,e1_5_1_1__funct_2]),
    [interesting(0.8),file(funct_2,e2_5__funct_2),[file(funct_2,e2_5__funct_2)]]).

fof(e3_5__funct_2,plain,
    ( v1_funct_1(c1_5__funct_2)
    & v1_funct_2(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2))
    & m2_relset_1(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__funct_2])],[rc3_relat_1,rc4_funct_1,antisymmetry_r2_hidden,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_partfun1,rc3_funct_1,t1_subset,t4_subset,t5_subset,reflexivity_r1_tarski,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc4_subset_1,fc5_relat_1,fc6_relat_1,fc7_relat_1,fc8_relat_1,rc1_relat_1,rc1_subset_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_subset_1,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,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_relset_1,fc1_subset_1,rc1_funct_1,rc1_funct_2,rc2_partfun1,t3_subset,existence_m2_relset_1,redefinition_m2_relset_1,dt_k1_relat_1,dt_k2_relat_1,dt_m2_relset_1,dt_c1_5__funct_2,dt_c2_5__funct_2,de_c2_5__funct_2,e2_5__funct_2]),
    [interesting(0.8),file(funct_2,e3_5__funct_2),[file(funct_2,e3_5__funct_2)]]).

fof(i2_5__funct_2,theorem,(
    $true ),
    introduced(tautology,[file(funct_2,i2_5__funct_2)]),
    [interesting(0.8),trivial,file(funct_2,i2_5__funct_2)]).

fof(i1_5__funct_2,plain,
    ( v1_funct_1(c1_5__funct_2)
    & v1_funct_2(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2))
    & m2_relset_1(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__funct_2])],[e3_5__funct_2,i2_5__funct_2]),
    [interesting(0.8),file(funct_2,i1_5__funct_2),[file(funct_2,i1_5__funct_2)]]).

fof(i1_5_tmp__funct_2,plain,
    ( ( v1_relat_1(c1_5__funct_2)
      & v1_funct_1(c1_5__funct_2) )
   => ( v1_funct_1(c1_5__funct_2)
      & v1_funct_2(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2))
      & m2_relset_1(c1_5__funct_2,k1_relat_1(c1_5__funct_2),k2_relat_1(c1_5__funct_2)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__funct_2])],[dt_c1_5__funct_2,i1_5__funct_2]),
    [interesting(1),t3_funct_2]).

fof(t3_funct_2,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_funct_1(A)
        & v1_funct_2(A,k1_relat_1(A),k2_relat_1(A))
        & m2_relset_1(A,k1_relat_1(A),k2_relat_1(A)) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_5_tmp__funct_2,dh_c1_5__funct_2]),
    [interesting(1),file(funct_2,t3_funct_2),[file(funct_2,t3_funct_2)]]).