 But now note that since \(f,\phi_d\) and \(h_f\) are all total, the value \(\phi_d(h_f(d))\) is defined. Thus setting \(n = \phi_d(h_f(d))\) it follows from (\ref{lastrecthm2}) that \(\phi_n(y) \simeq \phi_{f(n)}(y)\) as desired.