Let $U$ be an open subset of $\Bbb{R}^{2}$ and
\begin{align*}\psi:U&\rightarrow\Bbb{R}^{4}\\ x\mapsto &(\psi_1(x),\psi_2(x),\psi_3(x),\psi_4(x)) \end{align*} a $\mathcal C^1$ function.
Assume that there exists $(\overline{x_1},\overline{x_2})\in U$ such that $\text{d}\psi_{(\overline{x_1},\overline{x_2})}$ is injective.
First, I needed to prove the existence of $i,j\in\{1,2,3,4\},$ $i<j$ and an open set $V\subset U$ containing $(\overline{x_1},\overline{x_2})$, such that, the function $h:V\rightarrow\Bbb{R}^{2}$ defined by
$$h(x_1,x_2)=(\psi_i(x_1,x_2),\psi_j(x_1,x_2))$$
is a diffeomorphism of class $\mathcal C^1$ into an open set $W\subset \Bbb{R}^{2}$.
I have proved the above statement using the inverse function theorem. Note that, if $\mathrm{d}\psi_{(\overline{x_1},\overline{x_2})}$ is injective, we can choose two rows $ i,j$ of this matrix that are linearly independent, so we can apply the inverse function theorem to $h$.
But the second question is where I'm stucked:
Consider $i=1,j=2$. I need to show that $\forall$ $(x_1,x_2)\in V,$ $ \exists!$ $(y_1,y_2)\in W$ and a function $f=(f_1,f_2):W\rightarrow \Bbb{R}^{2}$ such that $$\psi(x_1,x_2)=(y_1,y_2,f(y_1,y_2)).$$
My problem here is how to prove that $\dfrac{\partial \psi}{\partial x_3,x_4}$ is non-singular.
By the same idea of the first question, the linearly independents rows of $\mathrm{d}\psi_{(\overline{x_1},\overline{x_2})}$ are $1$ and $2$.
What can I do?
