% Mizar problem: t2_arytm_1,arytm_1,44,40 
fof(t2_arytm_1,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ( k8_arytm_2(A,B) = k1_xboole_0
              & A != k1_xboole_0
              & B != k1_xboole_0 ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k8_arytm_2,dt_k12_arytm_3,dt_k13_arytm_3,dt_k1_xboole_0,dt_k2_arytm_2,dt_k4_ordinal2,dt_k6_arytm_3,dt_k8_arytm_2,dt_m1_subset_1,existence_m1_subset_1,fc1_xboole_0,rc1_xboole_0,rc2_xboole_0,redefinition_k12_arytm_3,redefinition_k13_arytm_3,t13_arytm_2,t15_arytm_2,t16_arytm_2,t1_subset,t2_subset,t4_arytm_2]),
    [file(arytm_1,t2_arytm_1)]).

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

fof(commutativity_k8_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => k8_arytm_2(A,B) = k8_arytm_2(B,A) ) ),
    file(arytm_2,k8_arytm_2),
    []).

fof(dt_k12_arytm_3,axiom,
    ( v1_xboole_0(k12_arytm_3)
    & m1_subset_1(k12_arytm_3,k6_arytm_3) ),
    file(arytm_3,k12_arytm_3),
    []).

fof(dt_k13_arytm_3,axiom,
    ( ~ v1_xboole_0(k13_arytm_3)
    & v3_ordinal1(k13_arytm_3)
    & m1_subset_1(k13_arytm_3,k6_arytm_3) ),
    file(arytm_3,k13_arytm_3),
    []).

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

fof(dt_k2_arytm_2,axiom,(
    $true ),
    file(arytm_2,k2_arytm_2),
    []).

fof(dt_k4_ordinal2,axiom,
    ( v3_ordinal1(k4_ordinal2)
    & ~ v1_xboole_0(k4_ordinal2) ),
    file(ordinal2,k4_ordinal2),
    []).

fof(dt_k6_arytm_3,axiom,(
    $true ),
    file(arytm_3,k6_arytm_3),
    []).

fof(dt_k8_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => m1_subset_1(k8_arytm_2(A,B),k2_arytm_2) ) ),
    file(arytm_2,k8_arytm_2),
    []).

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(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(redefinition_k12_arytm_3,axiom,(
    k12_arytm_3 = k1_xboole_0 ),
    file(arytm_3,k12_arytm_3),
    []).

fof(redefinition_k13_arytm_3,axiom,(
    k13_arytm_3 = k4_ordinal2 ),
    file(arytm_3,k13_arytm_3),
    []).

fof(t13_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ! [C] : 
              ( m1_subset_1(C,k2_arytm_2)
             => k8_arytm_2(A,k8_arytm_2(B,C)) = k8_arytm_2(k8_arytm_2(A,B),C) ) ) ) ),
    file(arytm_2,t13_arytm_2),
    []).

fof(t15_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ~ ( A != k12_arytm_3
          & ! [B] : 
              ( m1_subset_1(B,k2_arytm_2)
             => k8_arytm_2(A,B) != k13_arytm_3 ) ) ) ),
    file(arytm_2,t15_arytm_2),
    []).

fof(t16_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k13_arytm_3
           => k8_arytm_2(A,B) = B ) ) ) ),
    file(arytm_2,t16_arytm_2),
    []).

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(t4_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k12_arytm_3
           => k8_arytm_2(A,B) = k12_arytm_3 ) ) ) ),
    file(arytm_2,t4_arytm_2),
    []).