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Let $M \subset \mathbb R^n$, $k \in \mathbb N$. Assume that $M$ is a $k$-dimensional manifold in $\mathbb R^n$, i.e. for each $x \in M$ there exists an open set $W \subset \mathbb R^k$ and a smooth function $f: W \rightarrow \mathbb R^n$ such that:

(1) $f(W)=M \cap U$,

(2) $f'(y)$ has rank $k$ for each $y \in W$,

(3) $f^{-1}: f(W)\rightarrow W$ is continuous.

($f$ is then called a parametrisation)

I wish to prove that for each $x \in M$

(4) there exist an open subset $U$ in $\mathbb R^n$ containing $x$, an open subset $V$ in $\mathbb R^n$ and diffeomorphism $h:U\rightarrow V$ such that

$h(U\cap M)=V\cap (\mathbb R^k \times \{0\}).$

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    Thanks. I found the proof in: M. Spivak, Calculus on manifolds, thr.5.22012-11-07

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