% Mizar ND problem: t1_arytm_1,arytm_1,21,25 
fof(dh_c1_1__arytm_1,definition,
    ( ( m1_subset_1(c1_1__arytm_1,k2_arytm_2)
     => ! [A] : 
          ( m1_subset_1(A,k2_arytm_2)
         => ( k7_arytm_2(c1_1__arytm_1,A) = A
           => c1_1__arytm_1 = k1_xboole_0 ) ) )
   => ! [B] : 
        ( m1_subset_1(B,k2_arytm_2)
       => ! [C] : 
            ( m1_subset_1(C,k2_arytm_2)
           => ( k7_arytm_2(B,C) = C
             => B = k1_xboole_0 ) ) ) ),
    introduced(definition,[new_symbol(c1_1__arytm_1),file(arytm_1,c1_1__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c1_1__arytm_1)]).

fof(dh_c2_1__arytm_1,definition,
    ( ( m1_subset_1(c2_1__arytm_1,k2_arytm_2)
     => ( k7_arytm_2(c1_1__arytm_1,c2_1__arytm_1) = c2_1__arytm_1
       => c1_1__arytm_1 = k1_xboole_0 ) )
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ( k7_arytm_2(c1_1__arytm_1,A) = A
         => c1_1__arytm_1 = k1_xboole_0 ) ) ),
    introduced(definition,[new_symbol(c2_1__arytm_1),file(arytm_1,c2_1__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c2_1__arytm_1)]).

fof(e2_1__arytm_1,assumption,(
    k7_arytm_2(c1_1__arytm_1,c2_1__arytm_1) = c2_1__arytm_1 ),
    introduced(assumption,[file(arytm_1,e2_1__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,e2_1__arytm_1)]).

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(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(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(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(commutativity_k7_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,k7_arytm_2)]).

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_k2_arytm_2,axiom,(
    $true ),
    file(arytm_2,k2_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,k2_arytm_2)]).

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),
    [interesting(0.9),axiom,file(arytm_2,k7_arytm_2)]).

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

fof(dt_c1_1__arytm_1,assumption,(
    m1_subset_1(c1_1__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c1_1__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c1_1__arytm_1)]).

fof(dt_c2_1__arytm_1,assumption,(
    m1_subset_1(c2_1__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c2_1__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c2_1__arytm_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(de_c3_1__arytm_1,definition,(
    c3_1__arytm_1 = k1_xboole_0 ),
    introduced(definition,[new_symbol(c3_1__arytm_1),file(arytm_1,c3_1__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c3_1__arytm_1)]).

fof(dt_k4_ordinal2,axiom,
    ( v3_ordinal1(k4_ordinal2)
    & ~ v1_xboole_0(k4_ordinal2) ),
    file(ordinal2,k4_ordinal2),
    [interesting(0.9),axiom,file(ordinal2,k4_ordinal2)]).

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

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(redefinition_k13_arytm_3,definition,(
    k13_arytm_3 = k4_ordinal2 ),
    file(arytm_3,k13_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k13_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_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),
    [interesting(0.9),axiom,file(arytm_3,k13_arytm_3)]).

fof(t21_arytm_2,theorem,
    ( r2_hidden(k12_arytm_3,k2_arytm_2)
    & r2_hidden(k13_arytm_3,k2_arytm_2) ),
    file(arytm_2,t21_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,t21_arytm_2)]).

fof(e1_1__arytm_1,plain,(
    m1_subset_1(k1_xboole_0,k2_arytm_2) ),
    inference(mizar_by,[status(thm),assumptions([])],[dt_k4_ordinal2,dt_k6_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,antisymmetry_r2_hidden,existence_m1_subset_1,redefinition_k12_arytm_3,redefinition_k13_arytm_3,dt_k12_arytm_3,dt_k13_arytm_3,dt_k1_xboole_0,dt_k2_arytm_2,dt_m1_subset_1,fc1_xboole_0,t1_subset,t21_arytm_2]),
    [interesting(0.8),file(arytm_1,e1_1__arytm_1),[file(arytm_1,e1_1__arytm_1)]]).

fof(dt_c3_1__arytm_1,plain,(
    m1_subset_1(c3_1__arytm_1,k2_arytm_2) ),
    inference(mizar_by,[status(thm),assumptions([])],[antisymmetry_r2_hidden,t1_subset,rc1_xboole_0,rc2_xboole_0,t2_subset,existence_m1_subset_1,dt_k1_xboole_0,dt_k2_arytm_2,dt_m1_subset_1,fc1_xboole_0,de_c3_1__arytm_1,e1_1__arytm_1]),
    [interesting(0.8),file(arytm_1,c3_1__arytm_1),[file(arytm_1,c3_1__arytm_1)]]).

fof(reflexivity_r1_tarski,theorem,(
    ! [A,B] : r1_tarski(A,A) ),
    file(tarski,r1_tarski),
    [interesting(0.9),axiom,file(tarski,r1_tarski)]).

fof(t4_subset,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ),
    file(subset,t4_subset),
    [interesting(0.9),axiom,file(subset,t4_subset)]).

fof(t5_subset,theorem,(
    ! [A,B,C] : 
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ),
    file(subset,t5_subset),
    [interesting(0.9),axiom,file(subset,t5_subset)]).

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),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(redefinition_m2_subset_1,definition,(
    ! [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),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

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),
    [interesting(0.9),axiom,file(arytm_2,k1_arytm_2)]).

fof(dt_k1_zfmisc_1,axiom,(
    $true ),
    file(zfmisc_1,k1_zfmisc_1),
    [interesting(0.9),axiom,file(zfmisc_1,k1_zfmisc_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),
    [interesting(0.9),axiom,file(subset_1,m2_subset_1)]).

fof(t3_subset,theorem,(
    ! [A,B] : 
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ),
    file(subset,t3_subset),
    [interesting(0.9),axiom,file(subset,t3_subset)]).

fof(commutativity_k5_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,k5_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),
    [interesting(0.9),axiom,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),
    [interesting(0.9),axiom,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),
    [interesting(0.9),axiom,file(arytm_2,k5_arytm_2)]).

fof(d8_arytm_2,definition,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,d8_arytm_2)]).

fof(e3_1__arytm_1,plain,(
    k7_arytm_2(c1_1__arytm_1,c2_1__arytm_1) = k7_arytm_2(c2_1__arytm_1,c3_1__arytm_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__arytm_1,dt_c2_1__arytm_1,e2_1__arytm_1])],[reflexivity_r1_tarski,antisymmetry_r2_hidden,t1_subset,t4_subset,t5_subset,existence_m2_subset_1,redefinition_m2_subset_1,dt_k1_arytm_2,dt_k1_xboole_0,dt_k1_zfmisc_1,dt_k6_arytm_3,dt_m2_subset_1,fc1_xboole_0,rc1_xboole_0,rc2_xboole_0,t2_subset,t3_subset,commutativity_k5_arytm_2,commutativity_k7_arytm_2,existence_m1_subset_1,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k2_arytm_2,dt_k3_arytm_2,dt_k4_arytm_2,dt_k5_arytm_2,dt_k7_arytm_2,dt_m1_subset_1,dt_c1_1__arytm_1,dt_c2_1__arytm_1,dt_c3_1__arytm_1,de_c3_1__arytm_1,e2_1__arytm_1,d8_arytm_2]),
    [interesting(0.8),file(arytm_1,e3_1__arytm_1),[file(arytm_1,e3_1__arytm_1)]]).

fof(t12_arytm_2,theorem,(
    ! [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,B) = k7_arytm_2(A,C)
               => B = C ) ) ) ) ),
    file(arytm_2,t12_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,t12_arytm_2)]).

fof(e4_1__arytm_1,plain,(
    c1_1__arytm_1 = k1_xboole_0 ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_1__arytm_1,dt_c2_1__arytm_1,e2_1__arytm_1])],[antisymmetry_r2_hidden,t1_subset,rc1_xboole_0,rc2_xboole_0,t2_subset,commutativity_k7_arytm_2,existence_m1_subset_1,dt_k1_xboole_0,dt_k2_arytm_2,dt_k7_arytm_2,dt_m1_subset_1,dt_c1_1__arytm_1,dt_c2_1__arytm_1,dt_c3_1__arytm_1,de_c3_1__arytm_1,fc1_xboole_0,e3_1__arytm_1,t12_arytm_2]),
    [interesting(0.8),file(arytm_1,e4_1__arytm_1),[file(arytm_1,e4_1__arytm_1)]]).

fof(i4_1__arytm_1,theorem,(
    $true ),
    introduced(tautology,[file(arytm_1,i4_1__arytm_1)]),
    [interesting(0.8),trivial,file(arytm_1,i4_1__arytm_1)]).

fof(i3_1__arytm_1,plain,(
    c1_1__arytm_1 = k1_xboole_0 ),
    inference(conclusion,[status(thm),assumptions([dt_c1_1__arytm_1,dt_c2_1__arytm_1,e2_1__arytm_1])],[e4_1__arytm_1,i4_1__arytm_1]),
    [interesting(0.8),file(arytm_1,i3_1__arytm_1),[file(arytm_1,i3_1__arytm_1)]]).

fof(i2_1__arytm_1,plain,
    ( k7_arytm_2(c1_1__arytm_1,c2_1__arytm_1) = c2_1__arytm_1
   => c1_1__arytm_1 = k1_xboole_0 ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_1__arytm_1,dt_c2_1__arytm_1]),discharge_asm(discharge,[e2_1__arytm_1])],[e2_1__arytm_1,i3_1__arytm_1]),
    [interesting(0.8),file(arytm_1,i2_1__arytm_1),[file(arytm_1,i2_1__arytm_1)]]).

fof(i2_1_tmp__arytm_1,plain,
    ( m1_subset_1(c2_1__arytm_1,k2_arytm_2)
   => ( k7_arytm_2(c1_1__arytm_1,c2_1__arytm_1) = c2_1__arytm_1
     => c1_1__arytm_1 = k1_xboole_0 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_1__arytm_1]),discharge_asm(discharge,[dt_c2_1__arytm_1])],[dt_c2_1__arytm_1,i2_1__arytm_1]),
    [interesting(0.8),i1_1__arytm_1]).

fof(i1_1__arytm_1,plain,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ( k7_arytm_2(c1_1__arytm_1,A) = A
       => c1_1__arytm_1 = k1_xboole_0 ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_1__arytm_1])],[i2_1_tmp__arytm_1,dh_c2_1__arytm_1]),
    [interesting(0.8),file(arytm_1,i1_1__arytm_1),[file(arytm_1,i1_1__arytm_1)]]).

fof(i1_1_tmp__arytm_1,plain,
    ( m1_subset_1(c1_1__arytm_1,k2_arytm_2)
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ( k7_arytm_2(c1_1__arytm_1,A) = A
         => c1_1__arytm_1 = k1_xboole_0 ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_1__arytm_1])],[dt_c1_1__arytm_1,i1_1__arytm_1]),
    [interesting(1),t1_arytm_1]).

fof(t1_arytm_1,theorem,(
    ! [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 ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_1_tmp__arytm_1,dh_c1_1__arytm_1]),
    [interesting(1),file(arytm_1,t1_arytm_1),[file(arytm_1,t1_arytm_1)]]).