Solution
= Solution
The total function $f:\mathbb N^k\to\mathbb N$ is Sigma-1 represented in $T$ when there is a <Sigma-1 formula> $F(\mathbf x,y)$ such that, for every standard tuple $\mathbf n$ and $m=f(\mathbf n)$,
$$
T\vdash\forall y\bigl(F(\overline{\mathbf n},y)\leftrightarrow y=\bar m\bigr).
$$
Thus $T$ proves the correct unique output on every standard input.