Inverse Function Theorem for Real Functions

Theorem
Let $n\geq1$ and $k\geq1$ be natural numbers.

Let $\Omega\subset \R^n$ be open.

$f : \Omega \to \R^n$ be a vector-valued function of class $C^k$.

Let $a\in\Omega$.

Let the differential $Df(a)$ of $f$ at $a$ be invertible.

Then there exist open sets $U\subset\Omega$ and $V\subset\R^n$ such that the restriction of $f$ to $U$ is a $C^k$-diffeomorphism $f:U\to V$.