  For a Nash submersion $\phi\colon X\to Y$, we study the complex
$\mathcal{SDR}(\phi)$ of Schwartz sections of the relative de Rham complex of
$\phi$. We define the notion of Schwartz sections of constructible sheaves on
Nash manifolds and prove that $\mathcal{SDR}(\phi)$ is homotopy equivalent to
the Schwartz sections of the proper push-forward $\phi_!\mathbb{R}_X$ of the
constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that
$\mathcal{SDR}(\phi)$ depends (up to homotopy equivalence) only on the homology
type of the map $\phi$. We also deduce that $\mathcal{SDR}(\phi)$ has Hausdorff
homology spaces.
