Conditions under which Commutative Semigroup is Group/Lemma 3

Lemma for Conditions under which Commutative Semigroup is Group
Suppose the following:

Then:
 * If $y \circ x = x$ and $z \circ w = w$, then $y = z$.

Proof
Then:

Thus:
 * $y \circ \paren {w \circ x} = w \circ x = z \circ \paren {w \circ x}$

and $y = z$ follows from Lemma 1.