  Kirszbraun's Theorem states that every Lipschitz map $S\to\mathbb R^n$, where
$S\subseteq \mathbb R^m$, has an extension to a Lipschitz map $\mathbb R^m \to
\mathbb R^n$ 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 prove versions of these theorems valid
for definable maps and sets in arbitrary definably complete expansions of
ordered fields.
