Chain Rule for Partial Derivatives

Theorem
Let $F: \R^2 \to \R$ be a real-valued function of $2$ variables.

Let $X: \R^2 \to \R$ and $Y: \R^2 \to \R$ also be real-valued functions of $2$ variables.

Let $F = \map f {x, y}$ be such that:

Then:
 * $F = \map F {u, v}$

and: