Product of Commuting Idempotent Elements is Idempotent

Theorem
Let $\struct {S, \circ}$ be a semigroup.

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

Let $a$ and $b$ commute:


 * $a \circ b = b \circ a$

Then $a \circ b$ is idempotent.

Proof
From we take it for granted that $\struct {S, \circ}$ is closed under $\circ$.

Hence:

Thus $a \circ b$ is idempotent.