Why Does the Constant Rank Theorem Hold True?

[center o bass]

There are two theorems from multivariable calculus that is very important for manifold theory.
The first is the inverse function theorem and the second is the "constant rank theorem". The latter states that

(Constant rank theorem). If $f : U\subset \mathbb{R}^n \to \mathbb{R}^m$ has constant rank $k$ in a neighborhood of a point $p \in U$ , then after a suitable change of coordinates near $p$ in $U$ and $f(p)$ in $\mathbb{R}^m$, the map $f$ assumes the form $(x^1,...,x^n)\mapsto (x^1,...,x^k,0,...,0)$.
More precisely, there are a diffeomorphism $G$ of a neighborhood of $p$ in $U$ sending $p$ to the origin in $\mathbb{R}^n$ and a diffeomorphism $F$ of a neighborhood of $f(p)$ in $\mathbb{R}^m$ sending $f(p)$ to the origin in $\mathbb{R}^m$ such that $(F ◦ f ◦ G)^{−1}(x^1,...,x^n) = (x^1,...,x^k,0,...,0).$

I've gone through the proof of the theorem, but I'm left with little intuition on why it has to be true. Therefore I wonder, do you have any intuitive explanation of why the theorem has to be true?

 

[micromass]

Very informally: it is true for linear maps for sure, this is easy to check. Smooth maps can then be locally approximated by linear maps. Since the smooth map has constant rank, the linear approximation doesn't change when going over to close-lying points.