Idempotent Elements form Subsemigroup of Commutative Semigroup

Theorem
Let $\struct {S, \circ}$ 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:


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

Then $\struct {I, \circ}$ is a subsemigroup of $\struct {S, \circ}$.