  We prove a definable version of the Whitney embedding theorem for
abstract-definable $\mathcal{C}^p$ manifolds with $1\leq p<\infty$, namely:
every abstract-definable $\mathcal{C}^p$ manifold is abstract-definable $C^p$
embedded into $R^N$, for some positive integer $N$. As a consequence, we show
that every abstract-definable $\mathcal{C}^p$ manifold has a compatible
$\mathcal{C}^{p+1}$ atlas.
