% Mizar ND problem: t5_xboole_1,xboole_1,65,38 
fof(dh_c1_5__xboole_1,definition,
    ( ! [A,B] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),B) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,B),k2_xboole_0(A,B))
   => ! [C,D,E] : k2_xboole_0(k2_xboole_0(C,D),E) = k2_xboole_0(k2_xboole_0(C,E),k2_xboole_0(D,E)) ),
    introduced(definition,[new_symbol(c1_5__xboole_1),file(xboole_1,c1_5__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c1_5__xboole_1)]).

fof(dh_c2_5__xboole_1,definition,
    ( ! [A] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),A) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),k2_xboole_0(c2_5__xboole_1,A))
   => ! [B,C] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,B),C) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,C),k2_xboole_0(B,C)) ),
    introduced(definition,[new_symbol(c2_5__xboole_1),file(xboole_1,c2_5__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c2_5__xboole_1)]).

fof(dh_c3_5__xboole_1,definition,
    ( k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),c3_5__xboole_1) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c3_5__xboole_1),k2_xboole_0(c2_5__xboole_1,c3_5__xboole_1))
   => ! [A] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),A) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),k2_xboole_0(c2_5__xboole_1,A)) ),
    introduced(definition,[new_symbol(c3_5__xboole_1),file(xboole_1,c3_5__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c3_5__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(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_5__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c1_5__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c1_5__xboole_1)]).

fof(dt_c2_5__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c2_5__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c2_5__xboole_1)]).

fof(dt_c3_5__xboole_1,assumption,(
    $true ),
    introduced(assumption,[file(xboole_1,c3_5__xboole_1)]),
    [interesting(0.8),axiom,file(xboole_1,c3_5__xboole_1)]).

fof(t4_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ),
    file(xboole_1,t4_xboole_1),
    [interesting(0.9),axiom,file(xboole_1,t4_xboole_1)]).

fof(e1_5_1__xboole_1,plain,(
    k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),c3_5__xboole_1) = k2_xboole_0(c1_5__xboole_1,k2_xboole_0(k2_xboole_0(c3_5__xboole_1,c3_5__xboole_1),c2_5__xboole_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1])],[commutativity_k2_xboole_0,idempotence_k2_xboole_0,dt_k2_xboole_0,dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1,t4_xboole_1]),
    [interesting(0.65),file(xboole_1,e1_5_1__xboole_1),[file(xboole_1,e1_5_1__xboole_1)]]).

fof(e2_5_1__xboole_1,plain,(
    k2_xboole_0(c1_5__xboole_1,k2_xboole_0(k2_xboole_0(c3_5__xboole_1,c3_5__xboole_1),c2_5__xboole_1)) = k2_xboole_0(c1_5__xboole_1,k2_xboole_0(c3_5__xboole_1,k2_xboole_0(c3_5__xboole_1,c2_5__xboole_1))) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1])],[commutativity_k2_xboole_0,idempotence_k2_xboole_0,dt_k2_xboole_0,dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1,t4_xboole_1]),
    [interesting(0.65),file(xboole_1,e2_5_1__xboole_1),[file(xboole_1,e2_5_1__xboole_1)]]).

fof(e3_5_1__xboole_1,plain,(
    k2_xboole_0(c1_5__xboole_1,k2_xboole_0(c3_5__xboole_1,k2_xboole_0(c3_5__xboole_1,c2_5__xboole_1))) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c3_5__xboole_1),k2_xboole_0(c2_5__xboole_1,c3_5__xboole_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1])],[commutativity_k2_xboole_0,idempotence_k2_xboole_0,dt_k2_xboole_0,dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1,t4_xboole_1]),
    [interesting(0.65),file(xboole_1,e3_5_1__xboole_1),[file(xboole_1,e3_5_1__xboole_1)]]).

fof(e1_5__xboole_1,plain,(
    k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),c3_5__xboole_1) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c3_5__xboole_1),k2_xboole_0(c2_5__xboole_1,c3_5__xboole_1)) ),
    inference(iterative_eq,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1])],[e1_5_1__xboole_1,e2_5_1__xboole_1,e3_5_1__xboole_1]),
    [interesting(0.8),file(xboole_1,e1_5__xboole_1),[file(xboole_1,e1_5__xboole_1)]]).

fof(e2_5__xboole_1,plain,(
    k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),c3_5__xboole_1) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c3_5__xboole_1),k2_xboole_0(c2_5__xboole_1,c3_5__xboole_1)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1])],[commutativity_k2_xboole_0,idempotence_k2_xboole_0,dt_k2_xboole_0,dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1,e1_5__xboole_1]),
    [interesting(0.8),file(xboole_1,e2_5__xboole_1),[file(xboole_1,e2_5__xboole_1)]]).

fof(i4_5__xboole_1,theorem,(
    $true ),
    introduced(tautology,[file(xboole_1,i4_5__xboole_1)]),
    [interesting(0.8),trivial,file(xboole_1,i4_5__xboole_1)]).

fof(i3_5__xboole_1,plain,(
    k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),c3_5__xboole_1) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c3_5__xboole_1),k2_xboole_0(c2_5__xboole_1,c3_5__xboole_1)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1,dt_c3_5__xboole_1])],[e2_5__xboole_1,i4_5__xboole_1]),
    [interesting(0.8),file(xboole_1,i3_5__xboole_1),[file(xboole_1,i3_5__xboole_1)]]).

fof(i3_5_tmp__xboole_1,plain,(
    k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),c3_5__xboole_1) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c3_5__xboole_1),k2_xboole_0(c2_5__xboole_1,c3_5__xboole_1)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1]),discharge_asm(discharge,[dt_c3_5__xboole_1])],[dt_c3_5__xboole_1,i3_5__xboole_1]),
    [interesting(0.8),i2_5__xboole_1]).

fof(i2_5__xboole_1,plain,(
    ! [A] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),A) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),k2_xboole_0(c2_5__xboole_1,A)) ),
    inference(let,[status(thm),assumptions([dt_c1_5__xboole_1,dt_c2_5__xboole_1])],[i3_5_tmp__xboole_1,dh_c3_5__xboole_1]),
    [interesting(0.8),file(xboole_1,i2_5__xboole_1),[file(xboole_1,i2_5__xboole_1)]]).

fof(i2_5_tmp__xboole_1,plain,(
    ! [A] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,c2_5__xboole_1),A) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),k2_xboole_0(c2_5__xboole_1,A)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_5__xboole_1]),discharge_asm(discharge,[dt_c2_5__xboole_1])],[dt_c2_5__xboole_1,i2_5__xboole_1]),
    [interesting(0.8),i1_5__xboole_1]).

fof(i1_5__xboole_1,plain,(
    ! [A,B] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),B) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,B),k2_xboole_0(A,B)) ),
    inference(let,[status(thm),assumptions([dt_c1_5__xboole_1])],[i2_5_tmp__xboole_1,dh_c2_5__xboole_1]),
    [interesting(0.8),file(xboole_1,i1_5__xboole_1),[file(xboole_1,i1_5__xboole_1)]]).

fof(i1_5_tmp__xboole_1,plain,(
    ! [A,B] : k2_xboole_0(k2_xboole_0(c1_5__xboole_1,A),B) = k2_xboole_0(k2_xboole_0(c1_5__xboole_1,B),k2_xboole_0(A,B)) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_5__xboole_1])],[dt_c1_5__xboole_1,i1_5__xboole_1]),
    [interesting(1),t5_xboole_1]).

fof(t5_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(k2_xboole_0(A,C),k2_xboole_0(B,C)) ),
    inference(let,[status(thm),assumptions([])],[i1_5_tmp__xboole_1,dh_c1_5__xboole_1]),
    [interesting(1),file(xboole_1,t5_xboole_1),[file(xboole_1,t5_xboole_1)]]).