Idempotent Semigroup/Properties/2

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

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

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

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

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