 The second part of  Proposition 3.8  follows by observing that if \(R\) is recursive then both \(R\) and \(\overline{R}\) are c.e. Thus if \(e\) is a c.e. index for \(R\), then \(\overline{R}\) is defined by \(\neg \exists z \tau_k(e,\vec{x},z)\) which is equivalent to a \(\Pi^0_1\)-formula since \(\tau_k(x,\vec{y},z) \in \Delta^0_1\).