Product of Commuting Idempotent Elements is Idempotent

Theorem
Let $\left({S, \circ}\right)$ be a semigroup.

Let $a, b \in S$ be idempotent elements of $S$.

Suppose that $a$ and $b$ commute.

That is, suppose that $a \circ b = b \circ a$.

Then $a \circ b$ is idempotent.

Proof
Thus $a \circ b$ is idempotent.