% Mizar ND problem: t3_arytm_1,arytm_1,56,32 
fof(dh_c1_4__arytm_1,definition,
    ( ( m1_subset_1(c1_4__arytm_1,k2_arytm_2)
     => ! [A] : 
          ( m1_subset_1(A,k2_arytm_2)
         => ! [B] : 
              ( m1_subset_1(B,k2_arytm_2)
             => ( ( r1_arytm_2(c1_4__arytm_1,A)
                  & r1_arytm_2(A,B) )
               => r1_arytm_2(c1_4__arytm_1,B) ) ) ) )
   => ! [C] : 
        ( m1_subset_1(C,k2_arytm_2)
       => ! [D] : 
            ( m1_subset_1(D,k2_arytm_2)
           => ! [E] : 
                ( m1_subset_1(E,k2_arytm_2)
               => ( ( r1_arytm_2(C,D)
                    & r1_arytm_2(D,E) )
                 => r1_arytm_2(C,E) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_4__arytm_1),file(arytm_1,c1_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c1_4__arytm_1)]).

fof(dh_c2_4__arytm_1,definition,
    ( ( m1_subset_1(c2_4__arytm_1,k2_arytm_2)
     => ! [A] : 
          ( m1_subset_1(A,k2_arytm_2)
         => ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
              & r1_arytm_2(c2_4__arytm_1,A) )
           => r1_arytm_2(c1_4__arytm_1,A) ) ) )
   => ! [B] : 
        ( m1_subset_1(B,k2_arytm_2)
       => ! [C] : 
            ( m1_subset_1(C,k2_arytm_2)
           => ( ( r1_arytm_2(c1_4__arytm_1,B)
                & r1_arytm_2(B,C) )
             => r1_arytm_2(c1_4__arytm_1,C) ) ) ) ),
    introduced(definition,[new_symbol(c2_4__arytm_1),file(arytm_1,c2_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c2_4__arytm_1)]).

fof(dh_c3_4__arytm_1,definition,
    ( ( m1_subset_1(c3_4__arytm_1,k2_arytm_2)
     => ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
          & r1_arytm_2(c2_4__arytm_1,c3_4__arytm_1) )
       => r1_arytm_2(c1_4__arytm_1,c3_4__arytm_1) ) )
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
            & r1_arytm_2(c2_4__arytm_1,A) )
         => r1_arytm_2(c1_4__arytm_1,A) ) ) ),
    introduced(definition,[new_symbol(c3_4__arytm_1),file(arytm_1,c3_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c3_4__arytm_1)]).

fof(e1_4__arytm_1,assumption,(
    r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1) ),
    introduced(assumption,[file(arytm_1,e1_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,e1_4__arytm_1)]).

fof(e4_4__arytm_1,assumption,(
    r1_arytm_2(c2_4__arytm_1,c3_4__arytm_1) ),
    introduced(assumption,[file(arytm_1,e4_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,e4_4__arytm_1)]).

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(connectedness_r1_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,r1_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_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_4__arytm_1,assumption,(
    m1_subset_1(c1_4__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c1_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c1_4__arytm_1)]).

fof(dt_c3_4__arytm_1,assumption,(
    m1_subset_1(c3_4__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c3_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c3_4__arytm_1)]).

fof(dh_c4_4__arytm_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & k7_arytm_2(c1_4__arytm_1,A) = c2_4__arytm_1 )
   => ( m1_subset_1(c4_4__arytm_1,k2_arytm_2)
      & k7_arytm_2(c1_4__arytm_1,c4_4__arytm_1) = c2_4__arytm_1 ) ),
    introduced(definition,[new_symbol(c4_4__arytm_1),file(arytm_1,c4_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c4_4__arytm_1)]).

fof(dt_c2_4__arytm_1,assumption,(
    m1_subset_1(c2_4__arytm_1,k2_arytm_2) ),
    introduced(assumption,[file(arytm_1,c2_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c2_4__arytm_1)]).

fof(t10_arytm_2,theorem,(
    ! [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),
    [interesting(0.9),axiom,file(arytm_2,t10_arytm_2)]).

fof(e2_4__arytm_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & k7_arytm_2(c1_4__arytm_1,A) = c2_4__arytm_1 ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__arytm_1,dt_c2_4__arytm_1,e1_4__arytm_1])],[commutativity_k7_arytm_2,connectedness_r1_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_k7_arytm_2,dt_m1_subset_1,dt_c1_4__arytm_1,dt_c2_4__arytm_1,e1_4__arytm_1,t10_arytm_2]),
    [interesting(0.8),file(arytm_1,e2_4__arytm_1),[file(arytm_1,e2_4__arytm_1)]]).

fof(dt_c4_4__arytm_1,plain,(
    m1_subset_1(c4_4__arytm_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c1_4__arytm_1,dt_c2_4__arytm_1,e1_4__arytm_1])],[dh_c4_4__arytm_1,e2_4__arytm_1]),
    [interesting(0.8),file(arytm_1,c4_4__arytm_1),[file(arytm_1,c4_4__arytm_1)]]).

fof(dh_c5_4__arytm_1,definition,
    ( ? [A] : 
        ( m1_subset_1(A,k2_arytm_2)
        & k7_arytm_2(c2_4__arytm_1,A) = c3_4__arytm_1 )
   => ( m1_subset_1(c5_4__arytm_1,k2_arytm_2)
      & k7_arytm_2(c2_4__arytm_1,c5_4__arytm_1) = c3_4__arytm_1 ) ),
    introduced(definition,[new_symbol(c5_4__arytm_1),file(arytm_1,c5_4__arytm_1)]),
    [interesting(0.8),axiom,file(arytm_1,c5_4__arytm_1)]).

fof(e5_4__arytm_1,plain,(
    ? [A] : 
      ( m1_subset_1(A,k2_arytm_2)
      & k7_arytm_2(c2_4__arytm_1,A) = c3_4__arytm_1 ) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1])],[commutativity_k7_arytm_2,connectedness_r1_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_k7_arytm_2,dt_m1_subset_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1,t10_arytm_2]),
    [interesting(0.8),file(arytm_1,e5_4__arytm_1),[file(arytm_1,e5_4__arytm_1)]]).

fof(dt_c5_4__arytm_1,plain,(
    m1_subset_1(c5_4__arytm_1,k2_arytm_2) ),
    inference(consider,[status(thm),assumptions([dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1])],[dh_c5_4__arytm_1,e5_4__arytm_1]),
    [interesting(0.8),file(arytm_1,c5_4__arytm_1),[file(arytm_1,c5_4__arytm_1)]]).

fof(e3_4__arytm_1,plain,(
    k7_arytm_2(c1_4__arytm_1,c4_4__arytm_1) = c2_4__arytm_1 ),
    inference(consider,[status(thm),assumptions([dt_c1_4__arytm_1,dt_c2_4__arytm_1,e1_4__arytm_1])],[dh_c4_4__arytm_1,e2_4__arytm_1]),
    [interesting(0.8),file(arytm_1,e3_4__arytm_1),[file(arytm_1,e3_4__arytm_1)]]).

fof(e6_4__arytm_1,plain,(
    k7_arytm_2(c2_4__arytm_1,c5_4__arytm_1) = c3_4__arytm_1 ),
    inference(consider,[status(thm),assumptions([dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1])],[dh_c5_4__arytm_1,e5_4__arytm_1]),
    [interesting(0.8),file(arytm_1,e6_4__arytm_1),[file(arytm_1,e6_4__arytm_1)]]).

fof(t7_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,k7_arytm_2(B,C)) = k7_arytm_2(k7_arytm_2(A,B),C) ) ) ) ),
    file(arytm_2,t7_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,t7_arytm_2)]).

fof(e7_4__arytm_1,plain,(
    c3_4__arytm_1 = k7_arytm_2(c1_4__arytm_1,k7_arytm_2(c4_4__arytm_1,c5_4__arytm_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__arytm_1,e1_4__arytm_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1])],[commutativity_k7_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_k7_arytm_2,dt_m1_subset_1,dt_c1_4__arytm_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1,dt_c4_4__arytm_1,dt_c5_4__arytm_1,e3_4__arytm_1,e6_4__arytm_1,t7_arytm_2]),
    [interesting(0.8),file(arytm_1,e7_4__arytm_1),[file(arytm_1,e7_4__arytm_1)]]).

fof(t20_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)
             => ( A = k7_arytm_2(B,C)
               => r1_arytm_2(C,A) ) ) ) ) ),
    file(arytm_2,t20_arytm_2),
    [interesting(0.9),axiom,file(arytm_2,t20_arytm_2)]).

fof(e8_4__arytm_1,plain,(
    r1_arytm_2(c1_4__arytm_1,c3_4__arytm_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_4__arytm_1,e1_4__arytm_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1])],[commutativity_k7_arytm_2,connectedness_r1_arytm_2,existence_m1_subset_1,dt_k2_arytm_2,dt_k7_arytm_2,dt_m1_subset_1,dt_c1_4__arytm_1,dt_c3_4__arytm_1,dt_c4_4__arytm_1,dt_c5_4__arytm_1,e7_4__arytm_1,t20_arytm_2]),
    [interesting(0.8),file(arytm_1,e8_4__arytm_1),[file(arytm_1,e8_4__arytm_1)]]).

fof(i6_4__arytm_1,theorem,(
    $true ),
    introduced(tautology,[file(arytm_1,i6_4__arytm_1)]),
    [interesting(0.8),trivial,file(arytm_1,i6_4__arytm_1)]).

fof(i5_4__arytm_1,plain,(
    r1_arytm_2(c1_4__arytm_1,c3_4__arytm_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_4__arytm_1,e1_4__arytm_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1,e4_4__arytm_1])],[e8_4__arytm_1,i6_4__arytm_1]),
    [interesting(0.8),file(arytm_1,i5_4__arytm_1),[file(arytm_1,i5_4__arytm_1)]]).

fof(i4_4__arytm_1,plain,(
    ~ ( r1_arytm_2(c2_4__arytm_1,c3_4__arytm_1)
      & ~ r1_arytm_2(c1_4__arytm_1,c3_4__arytm_1) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_4__arytm_1,e1_4__arytm_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1]),discharge_asm(discharge,[e4_4__arytm_1])],[e4_4__arytm_1,i5_4__arytm_1]),
    [interesting(0.8),file(arytm_1,i4_4__arytm_1),[file(arytm_1,i4_4__arytm_1)]]).

fof(i3_4__arytm_1,plain,
    ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
      & r1_arytm_2(c2_4__arytm_1,c3_4__arytm_1) )
   => r1_arytm_2(c1_4__arytm_1,c3_4__arytm_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_4__arytm_1,dt_c2_4__arytm_1,dt_c3_4__arytm_1]),discharge_asm(discharge,[e1_4__arytm_1])],[e1_4__arytm_1,i4_4__arytm_1]),
    [interesting(0.8),file(arytm_1,i3_4__arytm_1),[file(arytm_1,i3_4__arytm_1)]]).

fof(i3_4_tmp__arytm_1,plain,
    ( m1_subset_1(c3_4__arytm_1,k2_arytm_2)
   => ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
        & r1_arytm_2(c2_4__arytm_1,c3_4__arytm_1) )
     => r1_arytm_2(c1_4__arytm_1,c3_4__arytm_1) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_4__arytm_1,dt_c2_4__arytm_1]),discharge_asm(discharge,[dt_c3_4__arytm_1])],[dt_c3_4__arytm_1,i3_4__arytm_1]),
    [interesting(0.8),i2_4__arytm_1]).

fof(i2_4__arytm_1,plain,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
          & r1_arytm_2(c2_4__arytm_1,A) )
       => r1_arytm_2(c1_4__arytm_1,A) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_4__arytm_1,dt_c2_4__arytm_1])],[i3_4_tmp__arytm_1,dh_c3_4__arytm_1]),
    [interesting(0.8),file(arytm_1,i2_4__arytm_1),[file(arytm_1,i2_4__arytm_1)]]).

fof(i2_4_tmp__arytm_1,plain,
    ( m1_subset_1(c2_4__arytm_1,k2_arytm_2)
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ( ( r1_arytm_2(c1_4__arytm_1,c2_4__arytm_1)
            & r1_arytm_2(c2_4__arytm_1,A) )
         => r1_arytm_2(c1_4__arytm_1,A) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_4__arytm_1]),discharge_asm(discharge,[dt_c2_4__arytm_1])],[dt_c2_4__arytm_1,i2_4__arytm_1]),
    [interesting(0.8),i1_4__arytm_1]).

fof(i1_4__arytm_1,plain,(
    ! [A] : 
      ( m1_subset_1(A,k2_arytm_2)
     => ! [B] : 
          ( m1_subset_1(B,k2_arytm_2)
         => ( ( r1_arytm_2(c1_4__arytm_1,A)
              & r1_arytm_2(A,B) )
           => r1_arytm_2(c1_4__arytm_1,B) ) ) ) ),
    inference(let,[status(thm),assumptions([dt_c1_4__arytm_1])],[i2_4_tmp__arytm_1,dh_c2_4__arytm_1]),
    [interesting(0.8),file(arytm_1,i1_4__arytm_1),[file(arytm_1,i1_4__arytm_1)]]).

fof(i1_4_tmp__arytm_1,plain,
    ( m1_subset_1(c1_4__arytm_1,k2_arytm_2)
   => ! [A] : 
        ( m1_subset_1(A,k2_arytm_2)
       => ! [B] : 
            ( m1_subset_1(B,k2_arytm_2)
           => ( ( r1_arytm_2(c1_4__arytm_1,A)
                & r1_arytm_2(A,B) )
             => r1_arytm_2(c1_4__arytm_1,B) ) ) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_4__arytm_1])],[dt_c1_4__arytm_1,i1_4__arytm_1]),
    [interesting(1),t3_arytm_1]).

fof(t3_arytm_1,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)
             => ( ( r1_arytm_2(A,B)
                  & r1_arytm_2(B,C) )
               => r1_arytm_2(A,C) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_4_tmp__arytm_1,dh_c1_4__arytm_1]),
    [interesting(1),file(arytm_1,t3_arytm_1),[file(arytm_1,t3_arytm_1)]]).