% Mizar problem: t13_arytm_0,arytm_0,909,47 
fof(t13_arytm_0,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ( B = 0
           => k1_arytm_0(A,B) = A ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,cc1_arytm_3,cc2_arytm_3,cc4_arytm_3,commutativity_k1_arytm_0,commutativity_k2_xboole_0,commutativity_k5_arytm_2,commutativity_k7_arytm_2,d1_numbers,d1_tarski,d2_arytm_0,d2_xboole_0,d4_xboole_0,d8_arytm_2,dt_k12_arytm_3,dt_k13_arytm_3,dt_k1_arytm_0,dt_k1_arytm_2,dt_k1_numbers,dt_k1_tarski,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_arytm_1,dt_k2_arytm_2,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k3_arytm_2,dt_k4_arytm_2,dt_k4_ordinal2,dt_k4_tarski,dt_k4_xboole_0,dt_k5_arytm_2,dt_k6_arytm_3,dt_k7_arytm_2,dt_m1_subset_1,dt_m2_subset_1,existence_m1_subset_1,existence_m2_subset_1,fc1_arytm_2,fc1_arytm_3,fc1_numbers,fc1_subset_1,fc1_xboole_0,fc1_zfmisc_1,fc2_arytm_2,fc2_subset_1,fc2_xboole_0,fc3_xboole_0,fc4_subset_1,fc8_arytm_3,idempotence_k2_xboole_0,rc1_arytm_3,rc1_subset_1,rc1_xboole_0,rc2_arytm_3,rc2_subset_1,rc2_xboole_0,rc3_arytm_3,redefinition_k12_arytm_3,redefinition_k13_arytm_3,redefinition_m2_subset_1,reflexivity_r1_tarski,spc0_boole,t103_zfmisc_1,t19_arytm_1,t1_boole,t1_subset,t21_arytm_2,t2_subset,t3_arytm_0,t3_boole,t3_subset,t4_boole,t4_subset,t5_subset,t6_boole,t7_boole,t8_boole]),
    [file(arytm_0,t13_arytm_0)]).

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

fof(cc1_arytm_3,axiom,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( m1_subset_1(B,A)
         => ( v1_ordinal1(B)
            & v2_ordinal1(B)
            & v3_ordinal1(B) ) ) ) ),
    file(arytm_3,cc1_arytm_3),
    []).

fof(cc2_arytm_3,axiom,(
    ! [A] : 
      ( ( v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A) ) ) ),
    file(arytm_3,cc2_arytm_3),
    []).

fof(cc4_arytm_3,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k6_arytm_3)
     => ( v3_ordinal1(A)
       => ( v1_ordinal1(A)
          & v2_ordinal1(A)
          & v3_ordinal1(A)
          & v4_ordinal2(A) ) ) ) ),
    file(arytm_3,cc4_arytm_3),
    []).

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_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(commutativity_k5_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_arytm_2)
        & m1_subset_1(B,k1_arytm_2) )
     => k5_arytm_2(A,B) = k5_arytm_2(B,A) ) ),
    file(arytm_2,k5_arytm_2),
    []).

fof(commutativity_k7_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => k7_arytm_2(A,B) = k7_arytm_2(B,A) ) ),
    file(arytm_2,k7_arytm_2),
    []).

fof(d1_numbers,axiom,(
    k1_numbers = k4_xboole_0(k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)),k1_tarski(k4_tarski(0,0))) ),
    file(numbers,d1_numbers),
    []).

fof(d1_tarski,axiom,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    []).

fof(d2_arytm_0,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_numbers)
     => ! [B] : 
          ( m1_subset_1(B,k1_numbers)
         => ! [C] : 
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = E
                            & C = k7_arytm_2(D,E) ) ) ) )
                & ( ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = D
                            & B = k4_tarski(0,E)
                            & C = k2_arytm_1(D,E) ) ) ) )
                & ( ( r2_hidden(B,k2_arytm_2)
                    & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                 => ( C = k1_arytm_0(A,B)
                  <=> ? [D] : 
                        ( m1_subset_1(D,k2_arytm_2)
                        & ? [E] : 
                            ( m1_subset_1(E,k2_arytm_2)
                            & A = k4_tarski(0,D)
                            & B = E
                            & C = k2_arytm_1(E,D) ) ) ) )
                & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_arytm_2) )
                    & ~ ( r2_hidden(A,k2_arytm_2)
                        & r2_hidden(B,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                    & ~ ( r2_hidden(B,k2_arytm_2)
                        & r2_hidden(A,k2_zfmisc_1(k1_tarski(0),k2_arytm_2)) )
                    & ~ ( C = k1_arytm_0(A,B)
                      <=> ? [D] : 
                            ( m1_subset_1(D,k2_arytm_2)
                            & ? [E] : 
                                ( m1_subset_1(E,k2_arytm_2)
                                & A = k4_tarski(0,D)
                                & B = k4_tarski(0,E)
                                & C = k4_tarski(0,k7_arytm_2(D,E)) ) ) ) ) ) ) ) ) ),
    file(arytm_0,d2_arytm_0),
    []).

fof(d2_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    []).

fof(d4_xboole_0,axiom,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    []).

fof(d8_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( B = k12_arytm_3
             => k7_arytm_2(A,B) = A )
            & ( A = k12_arytm_3
             => k7_arytm_2(A,B) = B )
            & ~ ( B != k12_arytm_3
                & A != k12_arytm_3
                & k7_arytm_2(A,B) != k4_arytm_2(k5_arytm_2(k3_arytm_2(A),k3_arytm_2(B))) ) ) ) ) ),
    file(arytm_2,d8_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_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_arytm_2,axiom,(
    m1_subset_1(k1_arytm_2,k1_zfmisc_1(k1_zfmisc_1(k6_arytm_3))) ),
    file(arytm_2,k1_arytm_2),
    []).

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

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    []).

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

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    []).

fof(dt_k2_arytm_1,axiom,(
    $true ),
    file(arytm_1,k2_arytm_1),
    []).

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

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    []).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    []).

fof(dt_k3_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => m2_subset_1(k3_arytm_2(A),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) ),
    file(arytm_2,k3_arytm_2),
    []).

fof(dt_k4_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k1_arytm_2)
     => m1_subset_1(k4_arytm_2(A),k2_arytm_2) ) ),
    file(arytm_2,k4_arytm_2),
    []).

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

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    []).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    []).

fof(dt_k5_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k1_arytm_2)
        & m1_subset_1(B,k1_arytm_2) )
     => m2_subset_1(k5_arytm_2(A,B),k1_zfmisc_1(k6_arytm_3),k1_arytm_2) ) ),
    file(arytm_2,k5_arytm_2),
    []).

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

fof(dt_k7_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => m1_subset_1(k7_arytm_2(A,B),k2_arytm_2) ) ),
    file(arytm_2,k7_arytm_2),
    []).

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

fof(dt_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
         => m1_subset_1(C,A) ) ) ),
    file(subset_1,m2_subset_1),
    []).

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

fof(existence_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ? [C] : m2_subset_1(C,A,B) ) ),
    file(subset_1,m2_subset_1),
    []).

fof(fc1_arytm_2,axiom,(
    ~ v1_xboole_0(k1_arytm_2) ),
    file(arytm_2,fc1_arytm_2),
    []).

fof(fc1_arytm_3,axiom,
    ( ~ v1_xboole_0(k4_ordinal2)
    & v1_ordinal1(k4_ordinal2)
    & v2_ordinal1(k4_ordinal2)
    & v3_ordinal1(k4_ordinal2)
    & v4_ordinal2(k4_ordinal2) ),
    file(arytm_3,fc1_arytm_3),
    []).

fof(fc1_numbers,axiom,(
    ~ v1_xboole_0(k1_numbers) ),
    file(numbers,fc1_numbers),
    []).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ),
    file(subset_1,fc1_subset_1),
    []).

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

fof(fc1_zfmisc_1,axiom,(
    ! [A,B] : ~ v1_xboole_0(k4_tarski(A,B)) ),
    file(zfmisc_1,fc1_zfmisc_1),
    []).

fof(fc2_arytm_2,axiom,(
    ~ v1_xboole_0(k2_arytm_2) ),
    file(arytm_2,fc2_arytm_2),
    []).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    []).

fof(fc2_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    []).

fof(fc3_xboole_0,axiom,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    []).

fof(fc4_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    []).

fof(fc8_arytm_3,axiom,(
    ~ v1_xboole_0(k6_arytm_3) ),
    file(arytm_3,fc8_arytm_3),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(rc1_arytm_3,axiom,(
    ? [A] : 
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A) ) ),
    file(arytm_3,rc1_arytm_3),
    []).

fof(rc1_subset_1,axiom,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ? [B] : 
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ),
    file(subset_1,rc1_subset_1),
    []).

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

fof(rc2_arytm_3,axiom,(
    ? [A] : 
      ( m1_subset_1(A,k6_arytm_3)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A) ) ),
    file(arytm_3,rc2_arytm_3),
    []).

fof(rc2_subset_1,axiom,(
    ! [A] : 
    ? [B] : 
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ),
    file(subset_1,rc2_subset_1),
    []).

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

fof(rc3_arytm_3,axiom,(
    ? [A] : 
      ( m1_subset_1(A,k6_arytm_3)
      & v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A) ) ),
    file(arytm_3,rc3_arytm_3),
    []).

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(redefinition_m2_subset_1,axiom,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ! [C] : 
          ( m2_subset_1(C,A,B)
        <=> m1_subset_1(C,B) ) ) ),
    file(subset_1,m2_subset_1),
    []).

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

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

fof(t103_zfmisc_1,axiom,(
    ! [A,B,C,D] : ~ ( r1_tarski(A,k2_zfmisc_1(B,C))
      & r2_hidden(D,A)
      & ! [E,F] : ~ ( r2_hidden(E,B)
          & r2_hidden(F,C)
          & D = k4_tarski(E,F) ) ) ),
    file(zfmisc_1,t103_zfmisc_1),
    []).

fof(t19_arytm_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( A = k1_xboole_0
           => ( B = k1_xboole_0
              | k2_arytm_1(A,B) = k4_tarski(k1_xboole_0,B) ) ) ) ) ),
    file(arytm_1,t19_arytm_1),
    []).

fof(t1_boole,axiom,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    []).

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

fof(t21_arytm_2,axiom,
    ( r2_hidden(k12_arytm_3,k2_arytm_2)
    & r2_hidden(k13_arytm_3,k2_arytm_2) ),
    file(arytm_2,t21_arytm_2),
    []).

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

fof(t3_arytm_0,axiom,(
    ! [A] : ~ ( r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers)
      & A = k12_arytm_3 ) ),
    file(arytm_0,t3_arytm_0),
    []).

fof(t3_boole,axiom,(
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t3_boole),
    []).

fof(t3_subset,axiom,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    []).

fof(t4_boole,axiom,(
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ),
    file(boole,t4_boole),
    []).

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

fof(t5_subset,axiom,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & m1_subset_1(B,k1_zfmisc_1(C))
      & v1_xboole_0(C) ) ),
    file(subset,t5_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),
    []).