% Mizar ND problem: t4_algspec1,algspec1,121,21 
fof(dh_c1_4__algspec1,definition,
    ( ( ( v1_relat_1(c1_4__algspec1)
        & v1_funct_1(c1_4__algspec1) )
     => ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A) )
         => ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B) )
             => ! [C] : 
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C) )
                 => ( ( r1_partfun1(c1_4__algspec1,A)
                      & r1_partfun1(B,C) )
                   => r1_partfun1(k5_relat_1(B,c1_4__algspec1),k5_relat_1(C,A)) ) ) ) ) )
   => ! [D] : 
        ( ( v1_relat_1(D)
          & v1_funct_1(D) )
       => ! [E] : 
            ( ( v1_relat_1(E)
              & v1_funct_1(E) )
           => ! [F] : 
                ( ( v1_relat_1(F)
                  & v1_funct_1(F) )
               => ! [G] : 
                    ( ( v1_relat_1(G)
                      & v1_funct_1(G) )
                   => ( ( r1_partfun1(D,E)
                        & r1_partfun1(F,G) )
                     => r1_partfun1(k5_relat_1(F,D),k5_relat_1(G,E)) ) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_4__algspec1),file(algspec1,c1_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c1_4__algspec1)]).

fof(dh_c2_4__algspec1,definition,
    ( ( ( v1_relat_1(c2_4__algspec1)
        & v1_funct_1(c2_4__algspec1) )
     => ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A) )
         => ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B) )
             => ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
                  & r1_partfun1(A,B) )
               => r1_partfun1(k5_relat_1(A,c1_4__algspec1),k5_relat_1(B,c2_4__algspec1)) ) ) ) )
   => ! [C] : 
        ( ( v1_relat_1(C)
          & v1_funct_1(C) )
       => ! [D] : 
            ( ( v1_relat_1(D)
              & v1_funct_1(D) )
           => ! [E] : 
                ( ( v1_relat_1(E)
                  & v1_funct_1(E) )
               => ( ( r1_partfun1(c1_4__algspec1,C)
                    & r1_partfun1(D,E) )
                 => r1_partfun1(k5_relat_1(D,c1_4__algspec1),k5_relat_1(E,C)) ) ) ) ) ),
    introduced(definition,[new_symbol(c2_4__algspec1),file(algspec1,c2_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c2_4__algspec1)]).

fof(dh_c3_4__algspec1,definition,
    ( ( ( v1_relat_1(c3_4__algspec1)
        & v1_funct_1(c3_4__algspec1) )
     => ! [A] : 
          ( ( v1_relat_1(A)
            & v1_funct_1(A) )
         => ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
              & r1_partfun1(c3_4__algspec1,A) )
           => r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(A,c2_4__algspec1)) ) ) )
   => ! [B] : 
        ( ( v1_relat_1(B)
          & v1_funct_1(B) )
       => ! [C] : 
            ( ( v1_relat_1(C)
              & v1_funct_1(C) )
           => ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
                & r1_partfun1(B,C) )
             => r1_partfun1(k5_relat_1(B,c1_4__algspec1),k5_relat_1(C,c2_4__algspec1)) ) ) ) ),
    introduced(definition,[new_symbol(c3_4__algspec1),file(algspec1,c3_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c3_4__algspec1)]).

fof(dh_c4_4__algspec1,definition,
    ( ( ( v1_relat_1(c4_4__algspec1)
        & v1_funct_1(c4_4__algspec1) )
     => ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
          & r1_partfun1(c3_4__algspec1,c4_4__algspec1) )
       => r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(c4_4__algspec1,c2_4__algspec1)) ) )
   => ! [A] : 
        ( ( v1_relat_1(A)
          & v1_funct_1(A) )
       => ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
            & r1_partfun1(c3_4__algspec1,A) )
         => r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(A,c2_4__algspec1)) ) ) ),
    introduced(definition,[new_symbol(c4_4__algspec1),file(algspec1,c4_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c4_4__algspec1)]).

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(symmetry_r1_partfun1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_partfun1(A,B)
       => r1_partfun1(B,A) ) ) ),
    file(partfun1,r1_partfun1),
    [interesting(0.9),axiom,file(partfun1,r1_partfun1)]).

fof(reflexivity_r1_partfun1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => r1_partfun1(A,A) ) ),
    file(partfun1,r1_partfun1),
    [interesting(0.9),axiom,file(partfun1,r1_partfun1)]).

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_funct_1,axiom,(
    $true ),
    file(funct_1,k1_funct_1),
    [interesting(0.9),axiom,file(funct_1,k1_funct_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_k3_xboole_0,axiom,(
    $true ),
    file(xboole_0,k3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k3_xboole_0)]).

fof(dt_k5_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k5_relat_1(A,B)) ) ),
    file(relat_1,k5_relat_1),
    [interesting(0.9),axiom,file(relat_1,k5_relat_1)]).

fof(dt_c1_4__algspec1,assumption,
    ( v1_relat_1(c1_4__algspec1)
    & v1_funct_1(c1_4__algspec1) ),
    introduced(assumption,[file(algspec1,c1_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c1_4__algspec1)]).

fof(dt_c2_4__algspec1,assumption,
    ( v1_relat_1(c2_4__algspec1)
    & v1_funct_1(c2_4__algspec1) ),
    introduced(assumption,[file(algspec1,c2_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c2_4__algspec1)]).

fof(dt_c3_4__algspec1,assumption,
    ( v1_relat_1(c3_4__algspec1)
    & v1_funct_1(c3_4__algspec1) ),
    introduced(assumption,[file(algspec1,c3_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c3_4__algspec1)]).

fof(dt_c4_4__algspec1,assumption,
    ( v1_relat_1(c4_4__algspec1)
    & v1_funct_1(c4_4__algspec1) ),
    introduced(assumption,[file(algspec1,c4_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c4_4__algspec1)]).

fof(fc1_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_relat_1(k5_relat_1(A,B))
        & v1_funct_1(k5_relat_1(A,B)) ) ) ),
    file(funct_1,fc1_funct_1),
    [interesting(0.9),axiom,file(funct_1,fc1_funct_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(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(e1_4__algspec1,assumption,(
    ! [A] : 
      ( r2_hidden(A,k3_xboole_0(k1_relat_1(c1_4__algspec1),k1_relat_1(c2_4__algspec1)))
     => k1_funct_1(c1_4__algspec1,A) = k1_funct_1(c2_4__algspec1,A) ) ),
    introduced(assumption,[file(algspec1,e1_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,e1_4__algspec1)]).

fof(e2_4__algspec1,assumption,(
    ! [A] : 
      ( r2_hidden(A,k3_xboole_0(k1_relat_1(c3_4__algspec1),k1_relat_1(c4_4__algspec1)))
     => k1_funct_1(c3_4__algspec1,A) = k1_funct_1(c4_4__algspec1,A) ) ),
    introduced(assumption,[file(algspec1,e2_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,e2_4__algspec1)]).

fof(d6_partfun1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_partfun1(A,B)
          <=> ! [C] : 
                ( r2_hidden(C,k3_xboole_0(k1_relat_1(A),k1_relat_1(B)))
               => k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ),
    file(partfun1,d6_partfun1),
    [interesting(0.9),axiom,file(partfun1,d6_partfun1)]).

fof(dt_c5_4__algspec1,assumption,(
    $true ),
    introduced(assumption,[file(algspec1,c5_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c5_4__algspec1)]).

fof(dh_c5_4__algspec1,definition,
    ( ~ ( r2_hidden(c5_4__algspec1,k3_xboole_0(k1_relat_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1)),k1_relat_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1))))
        & k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),c5_4__algspec1) != k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),c5_4__algspec1) )
   => ! [A] : 
        ~ ( r2_hidden(A,k3_xboole_0(k1_relat_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1)),k1_relat_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1))))
          & k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),A) != k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),A) ) ),
    introduced(definition,[new_symbol(c5_4__algspec1),file(algspec1,c5_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,c5_4__algspec1)]).

fof(e3_4__algspec1,assumption,(
    r2_hidden(c5_4__algspec1,k3_xboole_0(k1_relat_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1)),k1_relat_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1)))) ),
    introduced(assumption,[file(algspec1,e3_4__algspec1)]),
    [interesting(0.8),axiom,file(algspec1,e3_4__algspec1)]).

fof(rc1_pboole,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(pboole,rc1_pboole),
    [interesting(0.9),axiom,file(pboole,rc1_pboole)]).

fof(rc2_catalg_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & ~ v3_relat_1(A)
      & v1_funct_1(A) ) ),
    file(catalg_1,rc2_catalg_1),
    [interesting(0.9),axiom,file(catalg_1,rc2_catalg_1)]).

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(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_circcomb,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v2_funct_1(A) ) ),
    file(circcomb,rc1_circcomb),
    [interesting(0.9),axiom,file(circcomb,rc1_circcomb)]).

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(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(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_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_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_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(fc10_relat_1,theorem,(
    ! [A,B] : 
      ( ( v1_xboole_0(A)
        & v1_relat_1(B) )
     => ( v1_xboole_0(k5_relat_1(B,A))
        & v1_relat_1(k5_relat_1(B,A)) ) ) ),
    file(relat_1,fc10_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc10_relat_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(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(fc9_relat_1,theorem,(
    ! [A,B] : 
      ( ( v1_xboole_0(A)
        & v1_relat_1(B) )
     => ( v1_xboole_0(k5_relat_1(A,B))
        & v1_relat_1(k5_relat_1(A,B)) ) ) ),
    file(relat_1,fc9_relat_1),
    [interesting(0.9),axiom,file(relat_1,fc9_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_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_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(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(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(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(e4_4__algspec1,plain,
    ( r2_hidden(c5_4__algspec1,k1_relat_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1)))
    & r2_hidden(c5_4__algspec1,k1_relat_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1])],[rc1_pboole,rc2_catalg_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_circcomb,rc3_funct_1,t2_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc10_relat_1,fc1_funct_1,fc1_relat_1,fc5_relat_1,fc7_relat_1,fc9_relat_1,rc1_funct_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,t2_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_relat_1,dt_k3_xboole_0,dt_k5_relat_1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,t1_subset,t7_boole,e3_4__algspec1,d3_xboole_0]),
    [interesting(0.8),file(algspec1,e4_4__algspec1),[file(algspec1,e4_4__algspec1)]]).

fof(t21_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(k5_relat_1(C,B)))
          <=> ( r2_hidden(A,k1_relat_1(C))
              & r2_hidden(k1_funct_1(C,A),k1_relat_1(B)) ) ) ) ) ),
    file(funct_1,t21_funct_1),
    [interesting(0.9),axiom,file(funct_1,t21_funct_1)]).

fof(e5_4__algspec1,plain,
    ( r2_hidden(c5_4__algspec1,k1_relat_1(c3_4__algspec1))
    & r2_hidden(k1_funct_1(c3_4__algspec1,c5_4__algspec1),k1_relat_1(c1_4__algspec1))
    & r2_hidden(c5_4__algspec1,k1_relat_1(c4_4__algspec1))
    & r2_hidden(k1_funct_1(c4_4__algspec1,c5_4__algspec1),k1_relat_1(c2_4__algspec1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1])],[rc1_pboole,rc2_catalg_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_circcomb,rc3_funct_1,t8_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc10_relat_1,fc5_relat_1,fc7_relat_1,fc9_relat_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,t2_subset,t6_boole,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k5_relat_1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,fc1_funct_1,rc1_funct_1,t1_subset,t7_boole,e4_4__algspec1,t21_funct_1]),
    [interesting(0.8),file(algspec1,e5_4__algspec1),[file(algspec1,e5_4__algspec1)]]).

fof(e6_4__algspec1,plain,(
    r2_hidden(c5_4__algspec1,k3_xboole_0(k1_relat_1(c3_4__algspec1),k1_relat_1(c4_4__algspec1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1])],[rc1_pboole,rc2_catalg_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_circcomb,rc3_funct_1,t2_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc1_relat_1,fc5_relat_1,fc7_relat_1,rc1_funct_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,t2_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k3_xboole_0,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,t1_subset,t7_boole,e5_4__algspec1,d3_xboole_0]),
    [interesting(0.8),file(algspec1,e6_4__algspec1),[file(algspec1,e6_4__algspec1)]]).

fof(e7_4__algspec1,plain,(
    k1_funct_1(c3_4__algspec1,c5_4__algspec1) = k1_funct_1(c4_4__algspec1,c5_4__algspec1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1,e2_4__algspec1])],[rc1_pboole,rc2_catalg_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_circcomb,rc3_funct_1,t2_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc1_relat_1,fc5_relat_1,fc7_relat_1,rc1_funct_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,t2_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k3_xboole_0,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,t1_subset,t7_boole,e6_4__algspec1,e2_4__algspec1]),
    [interesting(0.8),file(algspec1,e7_4__algspec1),[file(algspec1,e7_4__algspec1)]]).

fof(t23_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ),
    file(funct_1,t23_funct_1),
    [interesting(0.9),axiom,file(funct_1,t23_funct_1)]).

fof(e8_4__algspec1,plain,
    ( r2_hidden(k1_funct_1(c3_4__algspec1,c5_4__algspec1),k3_xboole_0(k1_relat_1(c1_4__algspec1),k1_relat_1(c2_4__algspec1)))
    & k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),c5_4__algspec1) = k1_funct_1(c1_4__algspec1,k1_funct_1(c3_4__algspec1,c5_4__algspec1))
    & k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),c5_4__algspec1) = k1_funct_1(c2_4__algspec1,k1_funct_1(c4_4__algspec1,c5_4__algspec1)) ),
    inference(mizar_by,[status(thm),assumptions([e2_4__algspec1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1])],[rc1_pboole,rc2_catalg_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_circcomb,rc3_funct_1,t2_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc10_relat_1,fc5_relat_1,fc7_relat_1,fc9_relat_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,t2_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k3_xboole_0,dt_k5_relat_1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,fc1_funct_1,fc1_relat_1,rc1_funct_1,t1_subset,t7_boole,e7_4__algspec1,e5_4__algspec1,t23_funct_1,d3_xboole_0]),
    [interesting(0.8),file(algspec1,e8_4__algspec1),[file(algspec1,e8_4__algspec1)]]).

fof(e9_4__algspec1,plain,(
    k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),c5_4__algspec1) = k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),c5_4__algspec1) ),
    inference(mizar_by,[status(thm),assumptions([e1_4__algspec1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1,e2_4__algspec1])],[rc1_pboole,rc2_catalg_1,rc3_relat_1,rc4_funct_1,dt_k1_xboole_0,fc12_relat_1,fc1_xboole_0,fc4_relat_1,rc1_circcomb,rc3_funct_1,t2_boole,existence_m1_subset_1,dt_m1_subset_1,cc1_funct_1,cc1_relat_1,cc2_funct_1,fc10_relat_1,fc1_funct_1,fc1_relat_1,fc5_relat_1,fc7_relat_1,fc9_relat_1,rc1_funct_1,rc1_relat_1,rc1_xboole_0,rc2_funct_1,rc2_relat_1,rc2_xboole_0,t2_subset,t6_boole,t8_boole,commutativity_k3_xboole_0,idempotence_k3_xboole_0,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k3_xboole_0,dt_k5_relat_1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,t1_subset,t7_boole,e8_4__algspec1,e1_4__algspec1,e7_4__algspec1]),
    [interesting(0.8),file(algspec1,e9_4__algspec1),[file(algspec1,e9_4__algspec1)]]).

fof(i5_4__algspec1,theorem,(
    $true ),
    introduced(tautology,[file(algspec1,i5_4__algspec1)]),
    [interesting(0.8),trivial,file(algspec1,i5_4__algspec1)]).

fof(i4_4__algspec1,plain,(
    k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),c5_4__algspec1) = k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),c5_4__algspec1) ),
    inference(conclusion,[status(thm),assumptions([e1_4__algspec1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e3_4__algspec1,e2_4__algspec1])],[e9_4__algspec1,i5_4__algspec1]),
    [interesting(0.8),file(algspec1,i4_4__algspec1),[file(algspec1,i4_4__algspec1)]]).

fof(i3_4__algspec1,plain,(
    ~ ( r2_hidden(c5_4__algspec1,k3_xboole_0(k1_relat_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1)),k1_relat_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1))))
      & k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),c5_4__algspec1) != k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),c5_4__algspec1) ) ),
    inference(discharge_asm,[status(thm),assumptions([e1_4__algspec1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,dt_c5_4__algspec1,e2_4__algspec1]),discharge_asm(discharge,[e3_4__algspec1])],[e3_4__algspec1,i4_4__algspec1]),
    [interesting(0.8),file(algspec1,i3_4__algspec1),[file(algspec1,i3_4__algspec1)]]).

fof(i3_4_tmp__algspec1,plain,(
    ~ ( r2_hidden(c5_4__algspec1,k3_xboole_0(k1_relat_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1)),k1_relat_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1))))
      & k1_funct_1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),c5_4__algspec1) != k1_funct_1(k5_relat_1(c4_4__algspec1,c2_4__algspec1),c5_4__algspec1) ) ),
    inference(discharge_asm,[status(thm),assumptions([e1_4__algspec1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,e2_4__algspec1]),discharge_asm(discharge,[dt_c5_4__algspec1])],[dt_c5_4__algspec1,i3_4__algspec1]),
    [interesting(0.8),i2_4__algspec1]).

fof(i2_4__algspec1,plain,(
    r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(c4_4__algspec1,c2_4__algspec1)) ),
    inference(let,[status(thm),assumptions([e1_4__algspec1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,e2_4__algspec1])],[i3_4_tmp__algspec1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,symmetry_r1_partfun1,reflexivity_r1_partfun1,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k3_xboole_0,dt_k5_relat_1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,fc1_funct_1,fc1_relat_1,rc1_funct_1,d6_partfun1,dh_c5_4__algspec1]),
    [interesting(0.8),file(algspec1,i2_4__algspec1),[file(algspec1,i2_4__algspec1)]]).

fof(i2_4_tmp__algspec1,plain,
    ( ( ! [A] : 
          ( r2_hidden(A,k3_xboole_0(k1_relat_1(c1_4__algspec1),k1_relat_1(c2_4__algspec1)))
         => k1_funct_1(c1_4__algspec1,A) = k1_funct_1(c2_4__algspec1,A) )
      & ! [B] : 
          ( r2_hidden(B,k3_xboole_0(k1_relat_1(c3_4__algspec1),k1_relat_1(c4_4__algspec1)))
         => k1_funct_1(c3_4__algspec1,B) = k1_funct_1(c4_4__algspec1,B) ) )
   => r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(c4_4__algspec1,c2_4__algspec1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1]),discharge_asm(discharge,[e1_4__algspec1,e2_4__algspec1])],[e1_4__algspec1,e2_4__algspec1,i2_4__algspec1]),
    [interesting(0.8),i1_4__algspec1]).

fof(i1_4__algspec1,plain,
    ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
      & r1_partfun1(c3_4__algspec1,c4_4__algspec1) )
   => r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(c4_4__algspec1,c2_4__algspec1)) ),
    inference(mizar_def_expansion,[status(thm),assumptions([dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1])],[i2_4_tmp__algspec1,d6_partfun1,commutativity_k3_xboole_0,idempotence_k3_xboole_0,symmetry_r1_partfun1,reflexivity_r1_partfun1,antisymmetry_r2_hidden,dt_k1_funct_1,dt_k1_relat_1,dt_k3_xboole_0,dt_k5_relat_1,dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,fc1_funct_1,fc1_relat_1,rc1_funct_1]),
    [interesting(0.8),file(algspec1,i1_4__algspec1),[file(algspec1,i1_4__algspec1)]]).

fof(i1_4_tmp__algspec1,plain,
    ( ( v1_relat_1(c1_4__algspec1)
      & v1_funct_1(c1_4__algspec1)
      & v1_relat_1(c2_4__algspec1)
      & v1_funct_1(c2_4__algspec1)
      & v1_relat_1(c3_4__algspec1)
      & v1_funct_1(c3_4__algspec1)
      & v1_relat_1(c4_4__algspec1)
      & v1_funct_1(c4_4__algspec1) )
   => ( ( r1_partfun1(c1_4__algspec1,c2_4__algspec1)
        & r1_partfun1(c3_4__algspec1,c4_4__algspec1) )
     => r1_partfun1(k5_relat_1(c3_4__algspec1,c1_4__algspec1),k5_relat_1(c4_4__algspec1,c2_4__algspec1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1])],[dt_c1_4__algspec1,dt_c2_4__algspec1,dt_c3_4__algspec1,dt_c4_4__algspec1,i1_4__algspec1]),
    [interesting(1),t4_algspec1]).

fof(t4_algspec1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ! [D] : 
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ( ( r1_partfun1(A,B)
                      & r1_partfun1(C,D) )
                   => r1_partfun1(k5_relat_1(C,A),k5_relat_1(D,B)) ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_4_tmp__algspec1,dh_c1_4__algspec1,dh_c2_4__algspec1,dh_c3_4__algspec1,dh_c4_4__algspec1]),
    [interesting(1),file(algspec1,t4_algspec1),[file(algspec1,t4_algspec1)]]).