Pointwise Multiplication on Rational-Valued Functions is Associative

Theorem
Let $S$ be a non-empty set. Let $f, g, h: S \to \Q$ be rational-valued functions.

Let $f \times g: S \to \Q$ denote the pointwise product of $f$ and $g$.

Then:
 * $\paren {f \times g} \times h = f \times \paren {g \times h}$

That is, pointwise multiplication on rational-valued functions is associative.