 The left-to-right direction is subsumed under  Proposition 3.3.  For the right-to-left direction, suppose that \(A = \textrm{dom}(\phi_i)\) and \(\overline{A} = \textrm{dom}(\phi_j)\). Then to decide \(n \in A\) we can perform an unbounded search for a computation sequence \(s\) such that either \(T_1(i,n,s)\) or \(T_1(j,n,s)\), accepting in the first case and rejecting in the second. Since \(A \cup \overline{A} = \mathbb{N}\), the search must always terminate and since \(A \cap \overline{A} = \emptyset\), the conditions are exclusive. Thus by again appealing to Church’s Thesis, \(A\) is computable.