Pointwise Multiplication is Associative

Theorem
Let $S$ be a non-empty set.

Let $\mathbb F$ be one of the standard number sets: $\Z, \Q, \R$ or $\C$.

Let $f, g, h: S \to \mathbb F$ be functions.

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

Then:
 * $\left({f \times g}\right) \times h = f \times \left({g \times h}\right)$

That is, pointwise multiplication is associative.

Specific Contexts
This result can be applied and proved in the context of the various standard number sets: