% Mizar ND problem: t9_xboole_1,xboole_1,99,31 
fof(dh_c1_9__xboole_1,definition,
    ( ! [A,B] : 
        ( r1_tarski(c1_9__xboole_1,A)
       => r1_tarski(k2_xboole_0(c1_9__xboole_1,B),k2_xboole_0(A,B)) )
   => ! [C,D,E] : 
        ( r1_tarski(C,D)
       => r1_tarski(k2_xboole_0(C,E),k2_xboole_0(D,E)) ) ),
    introduced(definition,[new_symbol(c1_9__xboole_1),file(xboole_1,c1_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c1_9__xboole_1)]).

fof(dh_c2_9__xboole_1,definition,
    ( ! [A] : 
        ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
       => r1_tarski(k2_xboole_0(c1_9__xboole_1,A),k2_xboole_0(c2_9__xboole_1,A)) )
   => ! [B,C] : 
        ( r1_tarski(c1_9__xboole_1,B)
       => r1_tarski(k2_xboole_0(c1_9__xboole_1,C),k2_xboole_0(B,C)) ) ),
    introduced(definition,[new_symbol(c2_9__xboole_1),file(xboole_1,c2_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c2_9__xboole_1)]).

fof(dh_c3_9__xboole_1,definition,
    ( ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
     => r1_tarski(k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1),k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) )
   => ! [A] : 
        ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
       => r1_tarski(k2_xboole_0(c1_9__xboole_1,A),k2_xboole_0(c2_9__xboole_1,A)) ) ),
    introduced(definition,[new_symbol(c3_9__xboole_1),file(xboole_1,c3_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c3_9__xboole_1)]).

fof(e1_9__xboole_1,assumption,(
    r1_tarski(c1_9__xboole_1,c2_9__xboole_1) ),
    introduced(assumption,[file(xboole_1,e1_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,e1_9__xboole_1)]).

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

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

fof(dt_c1_9__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c1_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c1_9__xboole_1)]).

fof(dt_c2_9__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c2_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c2_9__xboole_1)]).

fof(dt_c3_9__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c3_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c3_9__xboole_1)]).

fof(dt_c4_9__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c4_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c4_9__xboole_1)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.9),axiom,file(tarski,d3_tarski)]).

fof(dh_c4_9__xboole_1,definition,
    ( ~ ( r2_hidden(c4_9__xboole_1,k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1))
        & ~ r2_hidden(c4_9__xboole_1,k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) )
   => ! [A] : 
        ~ ( r2_hidden(A,k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1))
          & ~ r2_hidden(A,k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ) ),
    introduced(definition,[new_symbol(c4_9__xboole_1),file(xboole_1,c4_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c4_9__xboole_1)]).

fof(e2_9__xboole_1,assumption,(
    r2_hidden(c4_9__xboole_1,k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1)) ),
    introduced(assumption,[file(xboole_1,e2_9__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,e2_9__xboole_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(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(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(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(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(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(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(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(e3_9__xboole_1,plain,
    ( r2_hidden(c4_9__xboole_1,c1_9__xboole_1)
    | r2_hidden(c4_9__xboole_1,c3_9__xboole_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,e2_9__xboole_1])],[dt_k1_xboole_0,fc1_xboole_0,t1_boole,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k2_xboole_0,dt_c1_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,t7_boole,e2_9__xboole_1,d2_xboole_0]),
    [interesting(0.8),file(xboole_1,e3_9__xboole_1),[file(xboole_1,e3_9__xboole_1)]]).

fof(e4_9__xboole_1,plain,
    ( r2_hidden(c4_9__xboole_1,c2_9__xboole_1)
    | r2_hidden(c4_9__xboole_1,c3_9__xboole_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,e2_9__xboole_1,e1_9__xboole_1])],[dt_k1_xboole_0,fc1_xboole_0,t8_boole,rc1_xboole_0,rc2_xboole_0,t6_boole,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_c1_9__xboole_1,dt_c2_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,t7_boole,e3_9__xboole_1,e1_9__xboole_1,d3_tarski]),
    [interesting(0.8),file(xboole_1,e4_9__xboole_1),[file(xboole_1,e4_9__xboole_1)]]).

fof(e5_9__xboole_1,plain,(
    r2_hidden(c4_9__xboole_1,k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,e2_9__xboole_1,e1_9__xboole_1])],[dt_k1_xboole_0,fc1_xboole_0,t1_boole,fc2_xboole_0,fc3_xboole_0,rc1_xboole_0,rc2_xboole_0,t6_boole,t8_boole,commutativity_k2_xboole_0,idempotence_k2_xboole_0,antisymmetry_r2_hidden,dt_k2_xboole_0,dt_c2_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,t7_boole,e4_9__xboole_1,d2_xboole_0]),
    [interesting(0.8),file(xboole_1,e5_9__xboole_1),[file(xboole_1,e5_9__xboole_1)]]).

fof(i7_9__xboole_1,theorem,(
    $true ),
    introduced(tautology,[file(xboole_1,i7_9__xboole_1)]),
    [interesting(0.8),trivial,file(xboole_1,i7_9__xboole_1)]).

fof(i6_9__xboole_1,plain,(
    r2_hidden(c4_9__xboole_1,k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,e2_9__xboole_1,e1_9__xboole_1])],[e5_9__xboole_1,i7_9__xboole_1]),
    [interesting(0.8),file(xboole_1,i6_9__xboole_1),[file(xboole_1,i6_9__xboole_1)]]).

fof(i5_9__xboole_1,plain,(
    ~ ( r2_hidden(c4_9__xboole_1,k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1))
      & ~ r2_hidden(c4_9__xboole_1,k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1,dt_c4_9__xboole_1,e1_9__xboole_1]),discharge_asm(discharge,[e2_9__xboole_1])],[e2_9__xboole_1,i6_9__xboole_1]),
    [interesting(0.8),file(xboole_1,i5_9__xboole_1),[file(xboole_1,i5_9__xboole_1)]]).

fof(i5_9_tmp__xboole_1,plain,(
    ~ ( r2_hidden(c4_9__xboole_1,k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1))
      & ~ r2_hidden(c4_9__xboole_1,k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1,e1_9__xboole_1]),discharge_asm(discharge,[dt_c4_9__xboole_1])],[dt_c4_9__xboole_1,i5_9__xboole_1]),
    [interesting(0.8),i4_9__xboole_1]).

fof(i4_9__xboole_1,plain,(
    r1_tarski(k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1),k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ),
    inference(let,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1,e1_9__xboole_1])],[i5_9_tmp__xboole_1,commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,antisymmetry_r2_hidden,dt_k2_xboole_0,dt_c1_9__xboole_1,dt_c2_9__xboole_1,dt_c3_9__xboole_1,d3_tarski,dh_c4_9__xboole_1]),
    [interesting(0.8),file(xboole_1,i4_9__xboole_1),[file(xboole_1,i4_9__xboole_1)]]).

fof(i3_9__xboole_1,plain,
    ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
   => r1_tarski(k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1),k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1,dt_c3_9__xboole_1]),discharge_asm(discharge,[e1_9__xboole_1])],[e1_9__xboole_1,i4_9__xboole_1]),
    [interesting(0.8),file(xboole_1,i3_9__xboole_1),[file(xboole_1,i3_9__xboole_1)]]).

fof(i3_9_tmp__xboole_1,plain,
    ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
   => r1_tarski(k2_xboole_0(c1_9__xboole_1,c3_9__xboole_1),k2_xboole_0(c2_9__xboole_1,c3_9__xboole_1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1]),discharge_asm(discharge,[dt_c3_9__xboole_1])],[dt_c3_9__xboole_1,i3_9__xboole_1]),
    [interesting(0.8),i2_9__xboole_1]).

fof(i2_9__xboole_1,plain,(
    ! [A] : 
      ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
     => r1_tarski(k2_xboole_0(c1_9__xboole_1,A),k2_xboole_0(c2_9__xboole_1,A)) ) ),
    inference(let,[status(thm),assumptions([dt_c2_9__xboole_1,dt_c1_9__xboole_1])],[i3_9_tmp__xboole_1,dh_c3_9__xboole_1]),
    [interesting(0.8),file(xboole_1,i2_9__xboole_1),[file(xboole_1,i2_9__xboole_1)]]).

fof(i2_9_tmp__xboole_1,plain,(
    ! [A] : 
      ( r1_tarski(c1_9__xboole_1,c2_9__xboole_1)
     => r1_tarski(k2_xboole_0(c1_9__xboole_1,A),k2_xboole_0(c2_9__xboole_1,A)) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_9__xboole_1]),discharge_asm(discharge,[dt_c2_9__xboole_1])],[dt_c2_9__xboole_1,i2_9__xboole_1]),
    [interesting(0.8),i1_9__xboole_1]).

fof(i1_9__xboole_1,plain,(
    ! [A,B] : 
      ( r1_tarski(c1_9__xboole_1,A)
     => r1_tarski(k2_xboole_0(c1_9__xboole_1,B),k2_xboole_0(A,B)) ) ),
    inference(let,[status(thm),assumptions([dt_c1_9__xboole_1])],[i2_9_tmp__xboole_1,dh_c2_9__xboole_1]),
    [interesting(0.8),file(xboole_1,i1_9__xboole_1),[file(xboole_1,i1_9__xboole_1)]]).

fof(i1_9_tmp__xboole_1,plain,(
    ! [A,B] : 
      ( r1_tarski(c1_9__xboole_1,A)
     => r1_tarski(k2_xboole_0(c1_9__xboole_1,B),k2_xboole_0(A,B)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_9__xboole_1])],[dt_c1_9__xboole_1,i1_9__xboole_1]),
    [interesting(1),t9_xboole_1]).

fof(t9_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => r1_tarski(k2_xboole_0(A,C),k2_xboole_0(B,C)) ) ),
    inference(let,[status(thm),assumptions([])],[i1_9_tmp__xboole_1,dh_c1_9__xboole_1]),
    [interesting(1),file(xboole_1,t9_xboole_1),[file(xboole_1,t9_xboole_1)]]).