Idempotent Elements form Subsemigroup of Commutative Semigroup/Proof 1

Proof
By Subsemigroup Closure Test we need only show that:


 * For all $x, y \in I$: $x \circ y \in I$.

That is:


 * $\paren {x \circ y} \circ \paren {x \circ y} = x \circ y$

We reason as follows:

Hence the result.