 As mentioned above, the Recursion Theorem may often be used to present compact proofs of results which would traditionally be described as involving self-reference. For instance, an immediate consequence is that for every \(f(x)\) there is an \(n\) such that \(W_n = W_{f(n)}\). To see this note that  Theorem 3.5  entails the existence of such an \(n\) such that \(\phi_n(x) \simeq \phi_{f(n)}\) for every computable \(f(x)\). But since the domains of the functions must then coincide, it follows that \(W_n = W_{f(n)}\).