% Mizar ND problem: t2_arytm_0,arytm_0,44,61 
fof(dh_c1_3__arytm_0,definition,
    ( ( m1_subset_1(c1_3__arytm_0,k2_arytm_2)
     => ( c1_3__arytm_0 != k12_arytm_3
       => r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_numbers) ) )
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ( A != k12_arytm_3
         => r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers) ) ) ),
    introduced(definition,[new_symbol(c1_3__arytm_0),file(arytm_0,c1_3__arytm_0)]),
    [interesting(0.8),axiom,file(arytm_0,c1_3__arytm_0)]).

fof(e1_3__arytm_0,assumption,(
    c1_3__arytm_0 != k12_arytm_3 ),
    introduced(assumption,[file(arytm_0,e1_3__arytm_0)]),
    [interesting(0.8),axiom,file(arytm_0,e1_3__arytm_0)]).

fof(existence_m1_subset_1,axiom,(
    ! [A] : 
    ? [B] : m1_subset_1(B,A) ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(dt_k1_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

fof(dt_k6_arytm_3,axiom,(
    $true ),
    file(arytm_3,k6_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k6_arytm_3)]).

fof(dt_m1_subset_1,axiom,(
    $true ),
    file(subset_1,m1_subset_1),
    [interesting(0.9),axiom,file(subset_1,m1_subset_1)]).

fof(fc1_xboole_0,theorem,(
    v1_xboole_0(k1_xboole_0) ),
    file(xboole_0,fc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc1_xboole_0)]).

fof(fc2_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) ),
    file(xboole_0,fc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc2_xboole_0)]).

fof(fc3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) ),
    file(xboole_0,fc3_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,fc3_xboole_0)]).

fof(fc4_subset_1,theorem,(
    ! [A,B] : 
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ~ v1_xboole_0(k2_zfmisc_1(A,B)) ) ),
    file(subset_1,fc4_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc4_subset_1)]).

fof(fc8_arytm_3,theorem,(
    ~ v1_xboole_0(k6_arytm_3) ),
    file(arytm_3,fc8_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,fc8_arytm_3)]).

fof(rc1_xboole_0,theorem,(
    ? [A] : v1_xboole_0(A) ),
    file(xboole_0,rc1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc1_xboole_0)]).

fof(rc2_xboole_0,theorem,(
    ? [A] : ~ v1_xboole_0(A) ),
    file(xboole_0,rc2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,rc2_xboole_0)]).

fof(t1_boole,theorem,(
    ! [A] : k2_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t1_boole),
    [interesting(0.9),axiom,file(boole,t1_boole)]).

fof(t2_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ),
    file(subset,t2_subset),
    [interesting(0.9),axiom,file(subset,t2_subset)]).

fof(t3_boole,theorem,(
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ),
    file(boole,t3_boole),
    [interesting(0.9),axiom,file(boole,t3_boole)]).

fof(t4_boole,theorem,(
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ),
    file(boole,t4_boole),
    [interesting(0.9),axiom,file(boole,t4_boole)]).

fof(t6_boole,theorem,(
    ! [A] : 
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ),
    file(boole,t6_boole),
    [interesting(0.9),axiom,file(boole,t6_boole)]).

fof(t8_boole,theorem,(
    ! [A,B] : 
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ),
    file(boole,t8_boole),
    [interesting(0.9),axiom,file(boole,t8_boole)]).

fof(commutativity_k2_xboole_0,theorem,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(idempotence_k2_xboole_0,theorem,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(antisymmetry_r2_hidden,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ),
    file(hidden,r2_hidden),
    [interesting(0.9),axiom,file(hidden,r2_hidden)]).

fof(redefinition_k12_arytm_3,definition,(
    k12_arytm_3 = k1_xboole_0 ),
    file(arytm_3,k12_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k12_arytm_3)]).

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),
    [interesting(0.9),axiom,file(arytm_3,k12_arytm_3)]).

fof(dt_k1_numbers,axiom,(
    $true ),
    file(numbers,k1_numbers),
    [interesting(0.9),axiom,file(numbers,k1_numbers)]).

fof(dt_k1_tarski,axiom,(
    $true ),
    file(tarski,k1_tarski),
    [interesting(0.9),axiom,file(tarski,k1_tarski)]).

fof(dt_k2_arytm_2,axiom,(
    $true ),
    file(arytm_2,k2_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,k2_arytm_2)]).

fof(dt_k2_xboole_0,axiom,(
    $true ),
    file(xboole_0,k2_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k2_xboole_0)]).

fof(dt_k2_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k2_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k2_zfmisc_1)]).

fof(dt_k4_tarski,axiom,(
    $true ),
    file(tarski,k4_tarski),
    [interesting(0.9),axiom,file(tarski,k4_tarski)]).

fof(dt_k4_xboole_0,axiom,(
    $true ),
    file(xboole_0,k4_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k4_xboole_0)]).

fof(dt_c1_3__arytm_0,assumption,(
    m1_subset_1(c1_3__arytm_0,k2_arytm_2) ),
    introduced(assumption,[file(arytm_0,c1_3__arytm_0)]),
    [interesting(0.8),axiom,file(arytm_0,c1_3__arytm_0)]).

fof(fc1_numbers,theorem,(
    ~ v1_xboole_0(k1_numbers) ),
    file(numbers,fc1_numbers),
    [interesting(0.9),axiom,file(numbers,fc1_numbers)]).

fof(fc1_zfmisc_1,theorem,(
    ! [A,B] : ~ v1_xboole_0(k4_tarski(A,B)) ),
    file(zfmisc_1,fc1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,fc1_zfmisc_1)]).

fof(fc2_arytm_2,theorem,(
    ~ v1_xboole_0(k2_arytm_2) ),
    file(arytm_2,fc2_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,fc2_arytm_2)]).

fof(fc2_subset_1,theorem,(
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ),
    file(subset_1,fc2_subset_1),
    [interesting(0.9),axiom,file(subset_1,fc2_subset_1)]).

fof(t1_subset,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ),
    file(subset,t1_subset),
    [interesting(0.9),axiom,file(subset,t1_subset)]).

fof(t7_boole,theorem,(
    ! [A,B] : 
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ),
    file(boole,t7_boole),
    [interesting(0.9),axiom,file(boole,t7_boole)]).

fof(spc0_boole,theorem,(
    v1_xboole_0(0) ),
    file(boole,spc0_boole),
    [interesting(0.9),axiom,file(boole,spc0_boole)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.9),axiom,file(tarski,d1_tarski)]).

fof(e3_3__arytm_0,plain,(
    r2_hidden(k12_arytm_3,k1_tarski(k12_arytm_3)) ),
    inference(mizar_by,[status(thm),assumptions([])],[existence_m1_subset_1,dt_k1_xboole_0,dt_k6_arytm_3,dt_m1_subset_1,fc1_xboole_0,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k1_tarski,fc2_subset_1,t1_subset,t7_boole,d1_tarski]),
    [interesting(0.8),file(arytm_0,e3_3__arytm_0),[file(arytm_0,e3_3__arytm_0)]]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,t106_zfmisc_1)]).

fof(e4_3__arytm_0,plain,(
    r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k2_zfmisc_1(k1_tarski(k12_arytm_3),k2_arytm_2)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__arytm_0])],[existence_m1_subset_1,dt_k1_xboole_0,dt_k6_arytm_3,dt_m1_subset_1,fc1_xboole_0,fc4_subset_1,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k1_tarski,dt_k2_arytm_2,dt_k2_zfmisc_1,dt_k4_tarski,dt_c1_3__arytm_0,fc1_zfmisc_1,fc2_arytm_2,fc2_subset_1,t1_subset,t7_boole,e3_3__arytm_0,t106_zfmisc_1]),
    [interesting(0.8),file(arytm_0,e4_3__arytm_0),[file(arytm_0,e4_3__arytm_0)]]).

fof(d2_xboole_0,definition,(
    ! [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),
    [interesting(0.9),axiom,file(xboole_0,d2_xboole_0)]).

fof(e5_3__arytm_0,plain,(
    r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(k12_arytm_3),k2_arytm_2))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__arytm_0])],[existence_m1_subset_1,dt_k1_xboole_0,dt_k6_arytm_3,dt_m1_subset_1,fc1_xboole_0,fc2_xboole_0,fc3_xboole_0,fc4_subset_1,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t1_boole,t2_subset,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k1_tarski,dt_k2_arytm_2,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k4_tarski,dt_c1_3__arytm_0,fc1_zfmisc_1,fc2_arytm_2,fc2_subset_1,t1_subset,t7_boole,e4_3__arytm_0,d2_xboole_0]),
    [interesting(0.8),file(arytm_0,e5_3__arytm_0),[file(arytm_0,e5_3__arytm_0)]]).

fof(e1_3_1__arytm_0,assumption,(
    r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_tarski(k4_tarski(k12_arytm_3,k12_arytm_3))) ),
    introduced(assumption,[file(arytm_0,e1_3_1__arytm_0)]),
    [interesting(0.65),axiom,file(arytm_0,e1_3_1__arytm_0)]).

fof(e2_3_1__arytm_0,plain,(
    k4_tarski(k12_arytm_3,c1_3__arytm_0) = k4_tarski(k12_arytm_3,k12_arytm_3) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__arytm_0,e1_3_1__arytm_0])],[existence_m1_subset_1,dt_k1_xboole_0,dt_k2_arytm_2,dt_k6_arytm_3,dt_m1_subset_1,fc1_xboole_0,fc2_arytm_2,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t8_boole,antisymmetry_r2_hidden,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k1_tarski,dt_k4_tarski,dt_c1_3__arytm_0,fc1_zfmisc_1,fc2_subset_1,t1_subset,t7_boole,e1_3_1__arytm_0,d1_tarski]),
    [interesting(0.65),file(arytm_0,e2_3_1__arytm_0),[file(arytm_0,e2_3_1__arytm_0)]]).

fof(t33_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( k4_tarski(A,B) = k4_tarski(C,D)
     => ( A = C
        & B = D ) ) ),
    file(zfmisc_1,t33_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,t33_zfmisc_1)]).

fof(e3_3_1__arytm_0,plain,(
    ~ $true ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__arytm_0,e1_3_1__arytm_0,e1_3__arytm_0])],[antisymmetry_r2_hidden,t1_subset,existence_m1_subset_1,dt_k1_xboole_0,dt_k2_arytm_2,dt_k6_arytm_3,dt_m1_subset_1,fc1_xboole_0,fc2_arytm_2,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,t6_boole,t7_boole,t8_boole,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k4_tarski,dt_c1_3__arytm_0,fc1_zfmisc_1,e2_3_1__arytm_0,e1_3__arytm_0,t33_zfmisc_1]),
    [interesting(0.65),file(arytm_0,e3_3_1__arytm_0),[file(arytm_0,e3_3_1__arytm_0)]]).

fof(i2_3_1__arytm_0,theorem,(
    $true ),
    introduced(tautology,[file(arytm_0,i2_3_1__arytm_0)]),
    [interesting(0.65),trivial,file(arytm_0,i2_3_1__arytm_0)]).

fof(i1_3_1__arytm_0,plain,(
    ~ $true ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__arytm_0,e1_3_1__arytm_0,e1_3__arytm_0])],[e3_3_1__arytm_0,i2_3_1__arytm_0]),
    [interesting(0.65),file(arytm_0,i1_3_1__arytm_0),[file(arytm_0,i1_3_1__arytm_0)]]).

fof(e2_3__arytm_0,plain,(
    ~ r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_tarski(k4_tarski(k12_arytm_3,k12_arytm_3))) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__arytm_0,e1_3__arytm_0]),discharge_asm(discharge,[e1_3_1__arytm_0])],[e1_3_1__arytm_0,i1_3_1__arytm_0]),
    [interesting(0.8),file(arytm_0,e2_3__arytm_0),[file(arytm_0,e2_3__arytm_0)]]).

fof(d1_numbers,definition,(
    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),
    [interesting(0.9),axiom,file(numbers,d1_numbers)]).

fof(d4_xboole_0,definition,(
    ! [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),
    [interesting(0.9),axiom,file(xboole_0,d4_xboole_0)]).

fof(e6_3__arytm_0,plain,(
    r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_numbers) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__arytm_0,e1_3__arytm_0])],[existence_m1_subset_1,dt_k1_xboole_0,dt_k6_arytm_3,dt_m1_subset_1,fc1_xboole_0,fc2_xboole_0,fc3_xboole_0,fc4_subset_1,fc8_arytm_3,rc1_xboole_0,rc2_xboole_0,t1_boole,t2_subset,t3_boole,t4_boole,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k1_numbers,dt_k1_tarski,dt_k2_arytm_2,dt_k2_xboole_0,dt_k2_zfmisc_1,dt_k4_tarski,dt_k4_xboole_0,dt_c1_3__arytm_0,fc1_numbers,fc1_zfmisc_1,fc2_arytm_2,fc2_subset_1,t1_subset,t7_boole,spc0_boole,e5_3__arytm_0,e2_3__arytm_0,d1_numbers,d4_xboole_0]),
    [interesting(0.8),file(arytm_0,e6_3__arytm_0),[file(arytm_0,e6_3__arytm_0)]]).

fof(i3_3__arytm_0,theorem,(
    $true ),
    introduced(tautology,[file(arytm_0,i3_3__arytm_0)]),
    [interesting(0.8),trivial,file(arytm_0,i3_3__arytm_0)]).

fof(i2_3__arytm_0,plain,(
    r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_numbers) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_3__arytm_0,e1_3__arytm_0])],[e6_3__arytm_0,i3_3__arytm_0]),
    [interesting(0.8),file(arytm_0,i2_3__arytm_0),[file(arytm_0,i2_3__arytm_0)]]).

fof(i1_3__arytm_0,plain,
    ( c1_3__arytm_0 != k12_arytm_3
   => r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_numbers) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__arytm_0]),discharge_asm(discharge,[e1_3__arytm_0])],[e1_3__arytm_0,i2_3__arytm_0]),
    [interesting(0.8),file(arytm_0,i1_3__arytm_0),[file(arytm_0,i1_3__arytm_0)]]).

fof(i1_3_tmp__arytm_0,plain,
    ( m1_subset_1(c1_3__arytm_0,k2_arytm_2)
   => ( c1_3__arytm_0 != k12_arytm_3
     => r2_hidden(k4_tarski(k12_arytm_3,c1_3__arytm_0),k1_numbers) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__arytm_0])],[dt_c1_3__arytm_0,i1_3__arytm_0]),
    [interesting(1),t2_arytm_0]).

fof(t2_arytm_0,theorem,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ( A != k12_arytm_3
       => r2_hidden(k4_tarski(k12_arytm_3,A),k1_numbers) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_3_tmp__arytm_0,dh_c1_3__arytm_0]),
    [interesting(1),file(arytm_0,t2_arytm_0),[file(arytm_0,t2_arytm_0)]]).