 An effective form of  Theorem 3.16  relating the \(\Delta^1_1\) subsets of \(\mathcal{C}\) to the lightface Borel sets representable by computable codes can be obtained from Kleene’s original proof of  Theorem 3.15  (see, e.g., Y. Moschovakis 2009, ch. 7B). Addison also showed that it is similarly possible to obtain an effective version of Lusin’s Theorem (1927)—i.e., “any two disjoint analytic sets can be separated by a Borel set”—and Kondô’s theorem (1939)—i.e., “every \(\mathbf{\Pi^1_1}\)-relation can be uniformized by a \(\mathbf{\Pi^1_1}\)-relation”. See Y. Moschovakis (2009, ch. 2E,4E) and also Simpson (2009, ch. V.3,VI.2)