Identity of Group is Unique/Proof 3

Proof
From Group has Latin Square Property, there exists a unique $x \in G$ such that:
 * $a x = b$

and there exists a unique $y \in G$ such that:
 * $y a = b$

Setting $b = a$, this becomes:

There exists a unique $x \in G$ such that:
 * $a x = a$

and there exists a unique $y \in G$ such that:
 * $y a = a$

These $x$ and $y$ are both $e$, by definition of identity element.