% Mizar problem: t38_relat_1,relat_1,677,23 
fof(t38_relat_1,conjecture,(
    ! [A] : 
      ( v1_relat_1(A)
     => k3_relat_1(A) = k3_relat_1(k4_relat_1(A)) ) ),
    inference(mizar_bg_added,[status(thm)],[commutativity_k2_xboole_0,d6_relat_1,dt_k1_relat_1,dt_k2_relat_1,dt_k2_xboole_0,dt_k3_relat_1,dt_k4_relat_1,fc2_relat_1,idempotence_k2_xboole_0,involutiveness_k4_relat_1,t37_relat_1]),
    [file(relat_1,t38_relat_1)]).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) ),
    file(xboole_0,k2_xboole_0),
    []).

fof(d6_relat_1,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => k3_relat_1(A) = k2_xboole_0(k1_relat_1(A),k2_relat_1(A)) ) ),
    file(relat_1,d6_relat_1),
    []).

fof(dt_k1_relat_1,axiom,(
    $true ),
    file(relat_1,k1_relat_1),
    []).

fof(dt_k2_relat_1,axiom,(
    $true ),
    file(relat_1,k2_relat_1),
    []).

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

fof(dt_k3_relat_1,axiom,(
    $true ),
    file(relat_1,k3_relat_1),
    []).

fof(dt_k4_relat_1,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => v1_relat_1(k4_relat_1(A)) ) ),
    file(relat_1,k4_relat_1),
    []).

fof(fc2_relat_1,axiom,(
    ! [A,B] : 
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k2_xboole_0(A,B)) ) ),
    file(relat_1,fc2_relat_1),
    []).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A ),
    file(xboole_0,k2_xboole_0),
    []).

fof(involutiveness_k4_relat_1,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => k4_relat_1(k4_relat_1(A)) = A ) ),
    file(relat_1,k4_relat_1),
    []).

fof(t37_relat_1,axiom,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( k2_relat_1(A) = k1_relat_1(k4_relat_1(A))
        & k1_relat_1(A) = k2_relat_1(k4_relat_1(A)) ) ) ),
    file(relat_1,t37_relat_1),
    []).