Composition of Commuting Idempotent Mappings is Idempotent

Theorem
Let $S$ be a set.

Let $f, g: S \to S$ be idempotent mappings from $S$ to $S$.

Let:
 * $f \circ g = g \circ f$

where $\circ$ denotes composition.

Then $f \circ g$ is idempotent.