Here we have $l > k$. I let $\phi : U \times \mathbb R^l \longrightarrow N(Z;\mathbb R^M)$ be the associated parametrization. I thought that to restrict $U \times \mathbb R^l$ to $U \times \mathbb R^k$ one should consider the projection map $\pi : U \times \mathbb R^l \longrightarrow U \times \mathbb R^k$. I'm not sure how that helps though, because I thought I needed to prove that the restricted parametrization $\pi\vert_{U \times \mathbb R^k}$ maps $U \times \mathbb R^k$ onto $N(Z;Y)$.