TUT CAL2 3

[!Theorem]
Let $U\subseteq {\mathbb{R}}^n$ be open. $\mathbf{f}:U\to {\mathbb{R}}^m$ be $C^1$ and $\mathbf{a}\in U$ such that $\text{rank}(\mathbf{Df}(\mathbf{a}))=r$. Show that there exist neighborhoods $V_1$ of $a$ (in $U$), $W_1$ of $f(a)$ , $V_2$ of $0$ in ${\mathbb{R}}^n$, and $W_2$ of $0$ in ${\mathbb{R}}^m$ along with $C_1$ diffeomorphisms $\phi :V_1\to V_2$ and $\psi :W_1\to W_2$ such that $\phi (a)=0$, and $\psi (\mathbf{f}(\mathbf{a}))=0$, and $\psi \circ f\circ {\phi }^{-1}(x_1,x_2,\dots ,x_n)=(x_1,\dots ,x_r,0,0,\dots ,0)$.