% Mizar ND problem: t11_xboole_1,xboole_1,118,29 
fof(dh_c1_11__xboole_1,definition,
    ( ! [A,B] : 
        ( r1_tarski(k2_xboole_0(c1_11__xboole_1,A),B)
       => r1_tarski(c1_11__xboole_1,B) )
   => ! [C,D,E] : 
        ( r1_tarski(k2_xboole_0(C,D),E)
       => r1_tarski(C,E) ) ),
    introduced(definition,[new_symbol(c1_11__xboole_1),file(xboole_1,c1_11__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c1_11__xboole_1)]).

fof(dh_c2_11__xboole_1,definition,
    ( ! [A] : 
        ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),A)
       => r1_tarski(c1_11__xboole_1,A) )
   => ! [B,C] : 
        ( r1_tarski(k2_xboole_0(c1_11__xboole_1,B),C)
       => r1_tarski(c1_11__xboole_1,C) ) ),
    introduced(definition,[new_symbol(c2_11__xboole_1),file(xboole_1,c2_11__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c2_11__xboole_1)]).

fof(dh_c3_11__xboole_1,definition,
    ( ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),c3_11__xboole_1)
     => r1_tarski(c1_11__xboole_1,c3_11__xboole_1) )
   => ! [A] : 
        ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),A)
       => r1_tarski(c1_11__xboole_1,A) ) ),
    introduced(definition,[new_symbol(c3_11__xboole_1),file(xboole_1,c3_11__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c3_11__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(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_11__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c1_11__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c1_11__xboole_1)]).

fof(dt_c2_11__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c2_11__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c2_11__xboole_1)]).

fof(dt_c3_11__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c3_11__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c3_11__xboole_1)]).

fof(t7_xboole_1,theorem,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    [interesting(0.9),axiom,file(xboole_1,t7_xboole_1)]).

fof(e1_11__xboole_1,plain,(
    r1_tarski(c1_11__xboole_1,k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_11__xboole_1,dt_c2_11__xboole_1])],[commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,dt_k2_xboole_0,dt_c1_11__xboole_1,dt_c2_11__xboole_1,t7_xboole_1]),
    [interesting(0.8),file(xboole_1,e1_11__xboole_1),[file(xboole_1,e1_11__xboole_1)]]).

fof(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    [interesting(0.9),axiom,file(xboole_1,t1_xboole_1)]).

fof(e2_11__xboole_1,plain,
    ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),c3_11__xboole_1)
   => r1_tarski(c1_11__xboole_1,c3_11__xboole_1) ),
    inference(mizar_by,[status(thm),assumptions([dt_c3_11__xboole_1,dt_c1_11__xboole_1,dt_c2_11__xboole_1])],[commutativity_k2_xboole_0,idempotence_k2_xboole_0,reflexivity_r1_tarski,dt_k2_xboole_0,dt_c1_11__xboole_1,dt_c2_11__xboole_1,dt_c3_11__xboole_1,e1_11__xboole_1,t1_xboole_1]),
    [interesting(0.8),file(xboole_1,e2_11__xboole_1),[file(xboole_1,e2_11__xboole_1)]]).

fof(i4_11__xboole_1,theorem,(
    $true ),
    introduced(tautology,[file(xboole_1,i4_11__xboole_1)]),
    [interesting(0.8),trivial,file(xboole_1,i4_11__xboole_1)]).

fof(i3_11__xboole_1,plain,
    ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),c3_11__xboole_1)
   => r1_tarski(c1_11__xboole_1,c3_11__xboole_1) ),
    inference(conclusion,[status(thm),assumptions([dt_c3_11__xboole_1,dt_c1_11__xboole_1,dt_c2_11__xboole_1])],[e2_11__xboole_1,i4_11__xboole_1]),
    [interesting(0.8),file(xboole_1,i3_11__xboole_1),[file(xboole_1,i3_11__xboole_1)]]).

fof(i3_11_tmp__xboole_1,plain,
    ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),c3_11__xboole_1)
   => r1_tarski(c1_11__xboole_1,c3_11__xboole_1) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_11__xboole_1,dt_c2_11__xboole_1]),discharge_asm(discharge,[dt_c3_11__xboole_1])],[dt_c3_11__xboole_1,i3_11__xboole_1]),
    [interesting(0.8),i2_11__xboole_1]).

fof(i2_11__xboole_1,plain,(
    ! [A] : 
      ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),A)
     => r1_tarski(c1_11__xboole_1,A) ) ),
    inference(let,[status(thm),assumptions([dt_c1_11__xboole_1,dt_c2_11__xboole_1])],[i3_11_tmp__xboole_1,dh_c3_11__xboole_1]),
    [interesting(0.8),file(xboole_1,i2_11__xboole_1),[file(xboole_1,i2_11__xboole_1)]]).

fof(i2_11_tmp__xboole_1,plain,(
    ! [A] : 
      ( r1_tarski(k2_xboole_0(c1_11__xboole_1,c2_11__xboole_1),A)
     => r1_tarski(c1_11__xboole_1,A) ) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_11__xboole_1]),discharge_asm(discharge,[dt_c2_11__xboole_1])],[dt_c2_11__xboole_1,i2_11__xboole_1]),
    [interesting(0.8),i1_11__xboole_1]).

fof(i1_11__xboole_1,plain,(
    ! [A,B] : 
      ( r1_tarski(k2_xboole_0(c1_11__xboole_1,A),B)
     => r1_tarski(c1_11__xboole_1,B) ) ),
    inference(let,[status(thm),assumptions([dt_c1_11__xboole_1])],[i2_11_tmp__xboole_1,dh_c2_11__xboole_1]),
    [interesting(0.8),file(xboole_1,i1_11__xboole_1),[file(xboole_1,i1_11__xboole_1)]]).

fof(i1_11_tmp__xboole_1,plain,(
    ! [A,B] : 
      ( r1_tarski(k2_xboole_0(c1_11__xboole_1,A),B)
     => r1_tarski(c1_11__xboole_1,B) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_11__xboole_1])],[dt_c1_11__xboole_1,i1_11__xboole_1]),
    [interesting(1),t11_xboole_1]).

fof(t11_xboole_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(k2_xboole_0(A,B),C)
     => r1_tarski(A,C) ) ),
    inference(let,[status(thm),assumptions([])],[i1_11_tmp__xboole_1,dh_c1_11__xboole_1]),
    [interesting(1),file(xboole_1,t11_xboole_1),[file(xboole_1,t11_xboole_1)]]).