 Next let \(k\) be an index for the composition of \(f(x)\) with \(b(x)\)—i.e., \(\phi_k(x) \simeq f(b(x))\). We now claim that \(n = b(k)\) is the number called for in the statement of the theorem. For first note that since \(b(x)\) and \(f(x)\) are both total, \(\phi_k(x)\) is also total and thus \(\phi_k(k)\) is defined. From this it follows that \(\phi_{b(k)}(y) \simeq \phi_{\phi_k(k)}(y)\). We now have the following sequence of functional identities: 