Idempotent Elements form Subsemigroup of Commutative Semigroup

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

Let $I$ be the set of all elements of $S$ that are idempotent under $\circ$.

That is:


 * $I = \left\{{x \in S: x \circ x = x}\right\}$

Then $\left({I, \circ}\right)$ is a subsemigroup of $\left({S, \circ}\right)$.