Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 12H / d / Solution

For each $x$, let $h(x)$ be an effectively obtained index of the partial functionThe existence of the total computable index transformation $h$ follows from the S-m-n theorem. By the recursion theorem, some $m$ satisfies $f_{m,1}\simeq f_{h(m),1}$. Therefore its domain isand hence