Idempotent Semigroup/Properties/1

Property of Idempotent Semigroup
Let $\struct {S, \circ}$ be an idempotent semigroup.

Let $x \circ y = y$ and $y \circ x = x$.

Then for all $z \in S$:
 * $z \circ x \circ z \circ y = z \circ y$

and:
 * $z \circ y \circ z \circ x = z \circ x$

Proof
From we take it for granted that $\circ$ is associative.