# A question on immersions

I am facing the following problem:

Let $\alpha:\mathbb{R}\rightarrow\mathbb{R}^2$ and $\beta:\mathbb{R}\rightarrow\mathbb{R}^2$ be $C^1$ curves with $\alpha(0)=(0,0)=\beta (0)$, such that $\alpha '(0)$ and $\beta '(0)$ are linearly independent. Show that there are open sets $U$ and $V$ in $\mathbb{R}^2$ and a $C^1$ diffeomorphism $\phi :U\rightarrow V$ such that $\phi(0,0)=(0,0)$, $\phi (\alpha(x))=(x,0)$ and $\phi (\beta (y))=(0,y)$ whenever $\alpha (x)$ and $\beta (y)$ are in $U$.

Using the Theorem for Local Form of Immersions (I don't know the english name for this theorem, I'm using a portuguese Analysis book and haven't found it elsewhere in the internet) I can find $C^1$ homeomorphisms $h_\alpha:\mathbb{R}^2\rightarrow\mathbb{R}^2$ and $h_\beta:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $h_\alpha\circ\alpha (t) = (t,0)$ and $h_\beta \circ \beta (t)=(0,t)$.

Now, I'm trying to define a function $h:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that it solves the problem; it should involve $h_\alpha$ and $h_\beta$ though I don't know how to assemble these parts together. Any help would be appreciated.

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Let $f:{\Bbb R}^2\to {\Bbb R}^2$ be $(x,y)\mapsto \alpha(x)+\beta(y)$. Then, use the inverse function theorem on $f$ at $(0,0)$.