The best proof of Clairaut's theorem on equality of mixed partials

This is the best proof of Clairaut's theorem on equality of mixed partials that I have ever thought up. And I bet it's the best proof of the theorem you'll ever see!

I'll prove the 2-variable version, but the generalization to $n$-variables is obvious.

Clairaut's Theorem: Suppose $f$ is a real-valued function of two variables $x,y$ and $f(x,y)$ is defined on an open subset $U$ of $\mathbb{R}^2$. Suppose further that both the second-order mixed partial derivatives $\partial_x\partial_y f(x,y)$ and $\partial_y\partial_x f(x,y)$ exist and are continuous on $U$. Then, we have: $$\partial_x\partial_y f = \partial_y\partial_x f$$ on all of $U$.