 Solution

Because the inclusion $\iota :N\hookrightarrow M$ has constant rank $n$, the constant rank theorem gives coordinates $y^1,\dots ,y^n$ on $N$ and $x^1,\dots ,x^m$ on $M$ in whichSince $\iota$ is an embedding, the ambient chart can be shrunk so that it meets no other local sheet of $N$. It then satisfiesand $(x^1,\dots ,x^n)$ restricts to the required chart on $N$. This is a slice chart for an embedded submanifold.