This is exercise 1.4.4 on Guillemin and Pollack's Differential Topology
Suppose that $Z \subset X \subset Y$ are manifolds, and $z \in Z$. Then there exist independent functions $g_1, \dots, g_l$, on a neighborhood $W$ of $z$ in $Y$ such that $$Z \cap W = \{y \in W : g_1(y) = 0, \dots, g_l(y) = 0\},$$ $$X \cap W = \{y \in W : g_i(y) = 0, \dots , g_m(y) = 0\},$$ where $l-m$ is the codimension of $Z$ in $X$.
I tried to set up the proof as following:
Suppose that $Z \subset X \subset Y$ are manifolds, and $z \in Z$. Let $Z$ and $X$ have codimensions $l$ and $m$ in $Y$, $Z$ has codimension $l-m$ in $X$. From the partial converse to the preimage theorem, there exist independent functions $f_1, \dots f_m$ on a neighborhood $U$ of $z$ in $Y$ such that $X \cap U$ is the common vanishing set of the $f_i$.
We also know that there exist independent functions $h_{m+1}, \dots, h_l$ on a neighborhood $V$ of $z$ in $X$ such that $Z \cap V$ is the common vanishing set of the $h_i$.
And then I don't know why $h_i$s are smooth, and how should I continue.
Any ideas? Thank you.