% Mizar problem: t17_arytm_0,arytm_0,1690,58 
fof(t17_arytm_0,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => k2_arytm_0(k3_arytm_0(A),B) = k3_arytm_0(k2_arytm_0(A,B)) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k1_arytm_0,commutativity_k2_arytm_0,d4_arytm_0,de_c2__arytm_0,dt_c2__arytm_0,dt_k1_arytm_0,dt_k1_numbers,dt_k1_xboole_0,dt_k2_arytm_0,dt_k3_arytm_0,dt_m1_subset_1,existence_m1_subset_1,fc1_numbers,fc1_xboole_0,involutiveness_k3_arytm_0,rc1_xboole_0,rc2_xboole_0,spc0_boole,t14_arytm_0,t16_arytm_0,t1_subset,t2_subset,t6_boole,t7_boole,t8_boole]),
    [file(arytm_0,t17_arytm_0)]).

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

fof(commutativity_k1_arytm_0,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k1_arytm_0(A,B) = k1_arytm_0(B,A) ) ),
    file(arytm_0,k1_arytm_0),
    []).

fof(commutativity_k2_arytm_0,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k2_arytm_0(A,B) = k2_arytm_0(B,A) ) ),
    file(arytm_0,k2_arytm_0),
    []).

fof(d4_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = k3_arytm_0(A)
          <=> k1_arytm_0(A,B) = 0 ) ) ) ),
    file(arytm_0,d4_arytm_0),
    []).

fof(de_c2__arytm_0,axiom,(
    c2__arytm_0 = 0 ),
    file(arytm_0,c2__arytm_0),
    []).

fof(dt_c2__arytm_0,axiom,(
    m1_subset_1(c2__arytm_0,k1_numbers) ),
    file(arytm_0,c2__arytm_0),
    []).

fof(dt_k1_arytm_0,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k1_arytm_0(A,B),k1_numbers) ) ),
    file(arytm_0,k1_arytm_0),
    []).

fof(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    []).

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

fof(dt_k2_arytm_0,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k2_arytm_0(A,B),k1_numbers) ) ),
    file(arytm_0,k2_arytm_0),
    []).

fof(dt_k3_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k3_arytm_0(A),k1_numbers) ) ),
    file(arytm_0,k3_arytm_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_numbers,axiom,(
    ~ v1_xboole_0(k1_numbers) ),
    file(numbers,fc1_numbers),
    []).

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

fof(involutiveness_k3_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => k3_arytm_0(k3_arytm_0(A)) = A ) ),
    file(arytm_0,k3_arytm_0),
    []).

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(t14_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = 0
           => k2_arytm_0(A,B) = 0 ) ) ) ),
    file(arytm_0,t14_arytm_0),
    []).

fof(t16_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => k2_arytm_0(A,k1_arytm_0(B,C)) = k1_arytm_0(k2_arytm_0(A,B),k2_arytm_0(A,C)) ) ) ) ),
    file(arytm_0,t16_arytm_0),
    []).

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),
    []).