Let $f:M^m\subset\mathbb R^k\to N^n$ be a smooth map between manifolds with $m>n$. Let $y\in N$ be a regular value for $f$. Let $x\in f^{-1}(y)$ and $L:\mathbb R^k\to\mathbb R^{m-n}$ be a linear map, non-singular on $R=\ker (df_x)$.
How can we see that the image of $f^{-1}(y)$ under $F=(f,L):M\to N\times\mathbb R^{m-n}$ is $y\times\mathbb R^{m-n}$?
$F$ is a diffeomorphism between a neighborhood $U\ni x$ and $V\ni y$. Can we see that $F$ is a diffeomorphism between $f^{-1}(y)\cap U$ and $y\times\mathbb R^{m-n}\cap V$?
Concluding that $f^{-1}(y)$ is a manifold follows.
(This related question adresses the first of the above issues but the answer is not convincing.)
