 \(f(x)\) thus treats its input as a pair \(\langle n,s \rangle\) consisting of an input \(n\) to \(\phi_e(x)\) and a computation sequence \(s\) as defined in the proof of the Normal Form Theorem  (2.3).  As \(x\) varies over \(\mathbb{N}\), it thus steps through all possible inputs \((x)_0\) to \(\phi_e\) and also all possible witnesses \((x)_1\) to the fact that the computation of \(\phi_e\) on \((x)_0\) halts. It then returns \((x)_0\) if \((x)_1\) is such a witness to a halting computation and \(a\) otherwise. Thus the range of \(f(x)\) will correspond to that of \(\phi_e(x)\). And as \(T_1(e,x,s)\) is computable (and in fact primitive recursive) relation, it is easy to see that \(f(x)\) is a computable function with range \(A\). This shows that \(A\) is c.e. as desired.