% Mizar problem: t4_arytm_1,arytm_1,69,30 
fof(t4_arytm_1,conjecture,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( r1_arytm_2(A,B)
              & r1_arytm_2(B,A) )
           => A = B ) ) ) ),
    inference(mizar_bg_added,[status(thm)],[antisymmetry_r2_hidden,commutativity_k5_arytm_2,commutativity_k7_arytm_2,connectedness_r1_arytm_2,d8_arytm_2,dt_k12_arytm_3,dt_k1_arytm_2,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k2_arytm_2,dt_k3_arytm_2,dt_k4_arytm_2,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_xboole_0,rc1_xboole_0,rc2_xboole_0,redefinition_k12_arytm_3,redefinition_m2_subset_1,reflexivity_r1_tarski,t10_arytm_2,t1_arytm_1,t1_subset,t2_subset,t3_subset,t4_subset,t5_subset,t6_arytm_2,t7_arytm_2]),
    [file(arytm_1,t4_arytm_1)]).

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

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(connectedness_r1_arytm_2,axiom,(
    ! [A,B] : 
      ( ( m1_subset_1(A,k2_arytm_2)
        & m1_subset_1(B,k2_arytm_2) )
     => ( r1_arytm_2(A,B)
        | r1_arytm_2(B,A) ) ) ),
    file(arytm_2,r1_arytm_2),
    []).

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_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_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_2,axiom,(
    $true ),
    file(arytm_2,k2_arytm_2),
    []).

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_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_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_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(t10_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ~ ( r1_arytm_2(A,B)
              & ! [C] : 
                  ( m1_subset_1(C,k2_arytm_2)
                 => k7_arytm_2(A,C) != B ) ) ) ) ),
    file(arytm_2,t10_arytm_2),
    []).

fof(t1_arytm_1,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( k7_arytm_2(A,B) = B
           => A = k1_xboole_0 ) ) ) ),
    file(arytm_1,t1_arytm_1),
    []).

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

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_arytm_2,axiom,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( k7_arytm_2(A,B) = k12_arytm_3
           => A = k12_arytm_3 ) ) ) ),
    file(arytm_2,t6_arytm_2),
    []).

fof(t7_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)
             => k7_arytm_2(A,k7_arytm_2(B,C)) = k7_arytm_2(k7_arytm_2(A,B),C) ) ) ) ),
    file(arytm_2,t7_arytm_2),
    []).