 In order to construct the formula \(\sigma_{k,1}(e,\vec{y})\) promised by  Theorem 3.9,  observe that standard techniques from the arithmetization of syntax allow us to obtain a \(\Delta^0_1\)-formula \(\tau_k(x,\vec{y},z)\) which defines the Kleene \(T\)-predicate \(T_k(x,\vec{y},z)\) introduced in  Section 2.2.2.  We may finally define \(\sigma_{k,1}(e,\vec{y}) = \exists z \tau_k(f(e),\vec{y},z)\). The first part of  Proposition 3.8  now follows by letting \(e\) be such that \(\textrm{dom}(\phi^k_e(\vec{x})) = R\) and then taking \(\sigma_{k,1}(e_0,\vec{x}) \in \Sigma^0_1\) where \(e_0\) is such that \(f(e_0) = e\). This is often formulated as what is known as the Enumeration Theorem which can be compared to  Theorem 3.2: