Kirszbraun's Theorem states that every Lipschitz map form $S$ to $\mathbb{R}^n$, where $S\subseteq \mathbb{R}^m$, has an extension to a Lipschitz map defined on $ämathbb{R^m}$ with the same Lipschitz constant. Its proof relies on Helly's Theorem: every family of compact subsets of $\mathbb{R}^n$, having the property that each of its subfamilies consisting of at most $n+1$ sets share a common point, has a non-empty intersection.

We sketch the proof of versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.