Inverse Function Theorem for Real Functions

Theorem
Let $n \ge 1$ and $k \ge 1$ be natural numbers.

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

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

Let $a \in \Omega$.

Let the differential $D \map f a$ of $f$ at $a$ be invertible.

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