% Mizar ND problem: t2_arytm_1,arytm_1,44,40 
fof(dh_c1_3__arytm_1,definition,
    ( ( m1_subset_1(c1_3__arytm_1,k2_arytm_2)
     => ! [A] : 
          ( m1_subset_1(A,k2_arytm_2)
         => ~ ( k8_arytm_2(c1_3__arytm_1,A) = k1_xboole_0
              & c1_3__arytm_1 != k1_xboole_0
              & A != k1_xboole_0 ) ) )
   => ! [B] : 
        ( m1_subset_1(B,k2_arytm_2)
       => ! [C] : 
            ( m1_subset_1(C,k2_arytm_2)
           => ~ ( k8_arytm_2(B,C) = k1_xboole_0
                & B != k1_xboole_0
                & C != k1_xboole_0 ) ) ) ),
    introduced(definition,[new_symbol(c1_3__arytm_1),file(arytm_1,c1_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c1_3__arytm_1)]).

fof(dh_c2_3__arytm_1,definition,
    ( ( m1_subset_1(c2_3__arytm_1,k2_arytm_2)
     => ~ ( k8_arytm_2(c1_3__arytm_1,c2_3__arytm_1) = k1_xboole_0
          & c1_3__arytm_1 != k1_xboole_0
          & c2_3__arytm_1 != k1_xboole_0 ) )
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ~ ( k8_arytm_2(c1_3__arytm_1,A) = k1_xboole_0
            & c1_3__arytm_1 != k1_xboole_0
            & A != k1_xboole_0 ) ) ),
    introduced(definition,[new_symbol(c2_3__arytm_1),file(arytm_1,c2_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c2_3__arytm_1)]).

fof(e1_3__arytm_1,assumption,(
    k8_arytm_2(c1_3__arytm_1,c2_3__arytm_1) = k1_xboole_0 ),
    introduced(assumption,[file(arytm_1,e1_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,e1_3__arytm_1)]).

fof(e2_3__arytm_1,assumption,(
    c1_3__arytm_1 != k1_xboole_0 ),
    introduced(assumption,[file(arytm_1,e2_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,e2_3__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(dt_k6_arytm_3,axiom,(
    $true ),
    file(arytm_3,k6_arytm_3),
    [interesting(0.9),axiom,file(arytm_3,k6_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(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_k8_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,k8_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(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_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(dt_k2_arytm_2,axiom,(
    $true ),
    file(arytm_2,k2_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,k2_arytm_2)]).

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_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),
    [interesting(0.9),axiom,file(arytm_2,k8_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_3__arytm_1,assumption,(
    m1_subset_1(c1_3__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c1_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c1_3__arytm_1)]).

fof(dt_c2_3__arytm_1,assumption,(
    m1_subset_1(c2_3__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c2_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c2_3__arytm_1)]).

fof(dh_c3_3__arytm_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & k8_arytm_2(c1_3__arytm_1,A) = k4_ordinal2 )
   => ( m1_subset_1(c3_3__arytm_1,k2_arytm_2)
      & k8_arytm_2(c1_3__arytm_1,c3_3__arytm_1) = k4_ordinal2 ) ),
    introduced(definition,[new_symbol(c3_3__arytm_1),file(arytm_1,c3_3__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c3_3__arytm_1)]).

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_xboole_0,axiom,(
    $true ),
    file(xboole_0,k1_xboole_0),
    [interesting(0.9),axiom,file(xboole_0,k1_xboole_0)]).

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(t15_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,t15_arytm_2)]).

fof(e3_3__arytm_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & k8_arytm_2(c1_3__arytm_1,A) = k4_ordinal2 ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_3__arytm_1,e2_3__arytm_1])],[antisymmetry_r2_hidden,t1_subset,dt_k6_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,commutativity_k8_arytm_2,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_k4_ordinal2,dt_k8_arytm_2,dt_m1_subset_1,dt_c1_3__arytm_1,fc1_xboole_0,e2_3__arytm_1,t15_arytm_2]),
    [interesting(0.8),file(arytm_1,e3_3__arytm_1),[file(arytm_1,e3_3__arytm_1)]]).

fof(dt_c3_3__arytm_1,plain,(
    m1_subset_1(c3_3__arytm_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_3__arytm_1,e2_3__arytm_1])],[dh_c3_3__arytm_1,e3_3__arytm_1]),
    [interesting(0.8),file(arytm_1,c3_3__arytm_1),[file(arytm_1,c3_3__arytm_1)]]).

fof(e4_3__arytm_1,plain,(
    k8_arytm_2(c1_3__arytm_1,c3_3__arytm_1) = k4_ordinal2 ),
    inference(consider,[status(thm),assumptions([dt_c1_3__arytm_1,e2_3__arytm_1])],[dh_c3_3__arytm_1,e3_3__arytm_1]),
    [interesting(0.8),file(arytm_1,e4_3__arytm_1),[file(arytm_1,e4_3__arytm_1)]]).

fof(t16_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,t16_arytm_2)]).

fof(e1_3_1__arytm_1,plain,(
    c2_3__arytm_1 = k8_arytm_2(k8_arytm_2(c1_3__arytm_1,c3_3__arytm_1),c2_3__arytm_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1,e2_3__arytm_1])],[antisymmetry_r2_hidden,t1_subset,dt_k6_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,commutativity_k8_arytm_2,existence_m1_subset_1,redefinition_k13_arytm_3,dt_k13_arytm_3,dt_k2_arytm_2,dt_k4_ordinal2,dt_k8_arytm_2,dt_m1_subset_1,dt_c1_3__arytm_1,dt_c2_3__arytm_1,dt_c3_3__arytm_1,e4_3__arytm_1,t16_arytm_2]),
    [interesting(0.65),file(arytm_1,e1_3_1__arytm_1),[file(arytm_1,e1_3_1__arytm_1)]]).

fof(t13_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)
             => k8_arytm_2(A,k8_arytm_2(B,C)) = k8_arytm_2(k8_arytm_2(A,B),C) ) ) ) ),
    file(arytm_2,t13_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,t13_arytm_2)]).

fof(e2_3_1__arytm_1,plain,(
    k8_arytm_2(k8_arytm_2(c1_3__arytm_1,c3_3__arytm_1),c2_3__arytm_1) = k8_arytm_2(k8_arytm_2(c1_3__arytm_1,c2_3__arytm_1),c3_3__arytm_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1,e2_3__arytm_1])],[commutativity_k8_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_k8_arytm_2,dt_m1_subset_1,dt_c1_3__arytm_1,dt_c2_3__arytm_1,dt_c3_3__arytm_1,t13_arytm_2]),
    [interesting(0.65),file(arytm_1,e2_3_1__arytm_1),[file(arytm_1,e2_3_1__arytm_1)]]).

fof(t4_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,t4_arytm_2)]).

fof(e3_3_1__arytm_1,plain,(
    k8_arytm_2(k8_arytm_2(c1_3__arytm_1,c2_3__arytm_1),c3_3__arytm_1) = k1_xboole_0 ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1,e2_3__arytm_1,e1_3__arytm_1])],[antisymmetry_r2_hidden,t1_subset,dt_k6_arytm_3,rc1_xboole_0,rc2_xboole_0,t2_subset,commutativity_k8_arytm_2,existence_m1_subset_1,redefinition_k12_arytm_3,dt_k12_arytm_3,dt_k1_xboole_0,dt_k2_arytm_2,dt_k8_arytm_2,dt_m1_subset_1,dt_c1_3__arytm_1,dt_c2_3__arytm_1,dt_c3_3__arytm_1,fc1_xboole_0,e1_3__arytm_1,t4_arytm_2]),
    [interesting(0.65),file(arytm_1,e3_3_1__arytm_1),[file(arytm_1,e3_3_1__arytm_1)]]).

fof(e5_3__arytm_1,plain,(
    c2_3__arytm_1 = k1_xboole_0 ),
    inference(iterative_eq,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1,e2_3__arytm_1,e1_3__arytm_1])],[e1_3_1__arytm_1,e2_3_1__arytm_1,e3_3_1__arytm_1]),
    [interesting(0.8),file(arytm_1,e5_3__arytm_1),[file(arytm_1,e5_3__arytm_1)]]).

fof(i5_3__arytm_1,theorem,(
    $true ),
    introduced(tautology,[file(arytm_1,i5_3__arytm_1)]),
    [interesting(0.8),trivial,file(arytm_1,i5_3__arytm_1)]).

fof(i4_3__arytm_1,plain,(
    c2_3__arytm_1 = k1_xboole_0 ),
    inference(conclusion,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1,e2_3__arytm_1,e1_3__arytm_1])],[e5_3__arytm_1,i5_3__arytm_1]),
    [interesting(0.8),file(arytm_1,i4_3__arytm_1),[file(arytm_1,i4_3__arytm_1)]]).

fof(i3_3__arytm_1,plain,(
    ~ ( c1_3__arytm_1 != k1_xboole_0
      & c2_3__arytm_1 != k1_xboole_0 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1,e1_3__arytm_1]),discharge_asm(discharge,[e2_3__arytm_1])],[e2_3__arytm_1,i4_3__arytm_1]),
    [interesting(0.8),file(arytm_1,i3_3__arytm_1),[file(arytm_1,i3_3__arytm_1)]]).

fof(i2_3__arytm_1,plain,(
    ~ ( k8_arytm_2(c1_3__arytm_1,c2_3__arytm_1) = k1_xboole_0
      & c1_3__arytm_1 != k1_xboole_0
      & c2_3__arytm_1 != k1_xboole_0 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_3__arytm_1,dt_c1_3__arytm_1]),discharge_asm(discharge,[e1_3__arytm_1])],[e1_3__arytm_1,i3_3__arytm_1]),
    [interesting(0.8),file(arytm_1,i2_3__arytm_1),[file(arytm_1,i2_3__arytm_1)]]).

fof(i2_3_tmp__arytm_1,plain,
    ( m1_subset_1(c2_3__arytm_1,k2_arytm_2)
   => ~ ( k8_arytm_2(c1_3__arytm_1,c2_3__arytm_1) = k1_xboole_0
        & c1_3__arytm_1 != k1_xboole_0
        & c2_3__arytm_1 != k1_xboole_0 ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_3__arytm_1]),discharge_asm(discharge,[dt_c2_3__arytm_1])],[dt_c2_3__arytm_1,i2_3__arytm_1]),
    [interesting(0.8),i1_3__arytm_1]).

fof(i1_3__arytm_1,plain,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ~ ( k8_arytm_2(c1_3__arytm_1,A) = k1_xboole_0
          & c1_3__arytm_1 != k1_xboole_0
          & A != k1_xboole_0 ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_3__arytm_1])],[i2_3_tmp__arytm_1,dh_c2_3__arytm_1]),
    [interesting(0.8),file(arytm_1,i1_3__arytm_1),[file(arytm_1,i1_3__arytm_1)]]).

fof(i1_3_tmp__arytm_1,plain,
    ( m1_subset_1(c1_3__arytm_1,k2_arytm_2)
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ~ ( k8_arytm_2(c1_3__arytm_1,A) = k1_xboole_0
            & c1_3__arytm_1 != k1_xboole_0
            & A != k1_xboole_0 ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_3__arytm_1])],[dt_c1_3__arytm_1,i1_3__arytm_1]),
    [interesting(1),t2_arytm_1]).

fof(t2_arytm_1,theorem,(
    ! [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(let,[status(thm),assumptions([])],[i1_3_tmp__arytm_1,dh_c1_3__arytm_1]),
    [interesting(1),file(arytm_1,t2_arytm_1),[file(arytm_1,t2_arytm_1)]]).