I have been reading about a stronger version of the Implicit Function Theorem which states the following:

Let $f(x,y) = (x_1,x_2,...,x_m, y_1,y_2,...,y_n)$ be a $C^{r+1}$ function from an open neighbourhood of $(a,b)$ $\in$ $\mathbb{R}^m \times \mathbb{R}^n$ $\to$ $\mathbb{R}^n$. Suppose that $f(a,b) = 0$ and $det(\frac{\partial{f}}{\partial{y}})(a,b) \neq 0$

Then there exists a $C^{r+1}$ function $y=g(x)$ on a neighbourhood of $a$, and a $C^r$ function $u(x,y)$ on a neighbourhood of $(a,b)$ with values in the space of $n\times n$ matrices, such that $g(a)=b$, $det\ u(x,y) \neq 0$ and

$f(x,y) = u(x,y) \bullet (y - g(x))$.

I was pretty fascinated by this theorem when I first read it but I cannot seem to find a proof of it and my attempts to prove it on my own have been unsuccessful.

Is anyone aware of a relatively easy to understand proof of this theorem?


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