% Mizar problem: t35_mod_2,mod_2,1013,37 
fof(t35_mod_2,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k1_enumset1(0,1,2))
     => ! [B] : 
          ( m1_subset_1(B,k1_enumset1(0,1,2))
         => ! [C] : 
              ( m1_subset_1(C,k1_enumset1(0,1,2))
             => ! [D] : 
                  ( m1_subset_1(D,k1_enumset1(0,1,2))
                 => ! [E] : 
                      ( m1_subset_1(E,k1_enumset1(0,1,2))
                     => ( ( D = 0
                          & E = 1 )
                       => ( k11_mod_2(A,B) = k11_mod_2(B,A)
                          & k11_mod_2(k11_mod_2(A,B),C) = k11_mod_2(A,k11_mod_2(B,C))
                          & k11_mod_2(A,D) = A
                          & k11_mod_2(A,k10_mod_2(A)) = D
                          & k12_mod_2(A,B) = k12_mod_2(B,A)
                          & k12_mod_2(k12_mod_2(A,B),C) = k12_mod_2(A,k12_mod_2(B,C))
                          & k12_mod_2(E,A) = A
                          & ~ ( A != D
                              & ! [F] : 
                                  ( m1_subset_1(F,k1_enumset1(0,1,2))
                                 => k12_mod_2(A,F) != E ) )
                          & D != E
                          & k12_mod_2(A,k11_mod_2(B,C)) = k11_mod_2(k12_mod_2(A,B),k12_mod_2(A,C)) ) ) ) ) ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,d16_mod_2,d17_mod_2,d1_enumset1,dt_k10_mod_2,dt_k11_mod_2,dt_k12_mod_2,dt_k1_enumset1,dt_k1_xboole_0,dt_m1_subset_1,existence_m1_subset_1,fc1_xboole_0,fc2_mod_2,l65_mod_2,rc1_xboole_0,rc2_xboole_0,spc0_boole,spc1_boole,spc2_boole,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole]),
    [file(mod_2,t35_mod_2)]).

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

fof(d16_mod_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_enumset1(0,1,2))
     => ! [B] : 
          ( m1_subset_1(B,k1_enumset1(0,1,2))
         => ( ( A = 0
             => k11_mod_2(A,B) = B )
            & ( B = 0
             => k11_mod_2(A,B) = A )
            & ( ( A = 1
                & B = 1 )
             => k11_mod_2(A,B) = 2 )
            & ( ( A = 1
                & B = 2 )
             => k11_mod_2(A,B) = 0 )
            & ( ( A = 2
                & B = 1 )
             => k11_mod_2(A,B) = 0 )
            & ( ( A = 2
                & B = 2 )
             => k11_mod_2(A,B) = 1 ) ) ) ) ),
    file(mod_2,d16_mod_2),
    []).

fof(d17_mod_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_enumset1(0,1,2))
     => ! [B] : 
          ( m1_subset_1(B,k1_enumset1(0,1,2))
         => ( ( B = 0
             => k12_mod_2(A,B) = 0 )
            & ( A = 0
             => k12_mod_2(A,B) = 0 )
            & ( B = 1
             => k12_mod_2(A,B) = A )
            & ( A = 1
             => k12_mod_2(A,B) = B )
            & ( ( A = 2
                & B = 2 )
             => k12_mod_2(A,B) = 1 ) ) ) ) ),
    file(mod_2,d17_mod_2),
    []).

fof(d1_enumset1,axiom,(
    ! [A,B,C,D] : 
      ( D = k1_enumset1(A,B,C)
    <=> ! [E] : 
          ( r2_hidden(E,D)
        <=> ~ ( E != A
              & E != B
              & E != C ) ) ) ),
    file(enumset1,d1_enumset1),
    []).

fof(dt_k10_mod_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_enumset1(0,1,2))
     => m1_subset_1(k10_mod_2(A),k1_enumset1(0,1,2)) ) ),
    file(mod_2,k10_mod_2),
    []).

fof(dt_k11_mod_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_enumset1(0,1,2))
        & m1_subset_1(B,k1_enumset1(0,1,2)) )
     => m1_subset_1(k11_mod_2(A,B),k1_enumset1(0,1,2)) ) ),
    file(mod_2,k11_mod_2),
    []).

fof(dt_k12_mod_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_enumset1(0,1,2))
        & m1_subset_1(B,k1_enumset1(0,1,2)) )
     => m1_subset_1(k12_mod_2(A,B),k1_enumset1(0,1,2)) ) ),
    file(mod_2,k12_mod_2),
    []).

fof(dt_k1_enumset1,axiom,(
    $true ),
    file(enumset1,k1_enumset1),
    []).

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

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    []).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    []).

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

fof(fc2_mod_2,axiom,(
    ! [A,B,C] : ~ v1_xboole_0(k1_enumset1(A,B,C)) ),
    file(mod_2,fc2_mod_2),
    []).

fof(l65_mod_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_enumset1(0,1,2))
     => ! [B] : 
          ( m1_subset_1(B,k1_enumset1(0,1,2))
         => ! [C] : 
              ( m1_subset_1(C,k1_enumset1(0,1,2))
             => ! [D] : 
                  ( m1_subset_1(D,k1_enumset1(0,1,2))
                 => ( D = 0
                   => ( k11_mod_2(A,k10_mod_2(A)) = D
                      & k11_mod_2(A,D) = A
                      & k11_mod_2(k11_mod_2(A,B),C) = k11_mod_2(A,k11_mod_2(B,C)) ) ) ) ) ) ) ),
    file(mod_2,l65_mod_2),
    []).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    []).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    []).

fof(spc0_boole,axiom,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    []).

fof(spc1_boole,axiom,(
    ~ v1_xboole_0(1) ),
    file(boole,spc1_boole),
    []).

fof(spc2_boole,axiom,(
    ~ v1_xboole_0(2) ),
    file(boole,spc2_boole),
    []).

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

fof(t2_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    []).

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

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

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