Semigroup is Group Iff Latin Square Property Holds/Proof 2

Theorem
Let $\left({S, \circ}\right)$ be a semigroup.

Then $\left({S, \circ}\right)$ is a group iff for all $a, b \in S$ the Latin square property holds in $S$:
 * $a \circ x = b$
 * $y \circ a = b$

for $x$ and $y$ each unique in $S$.

Necessary Condition
Let $\left({S, \circ}\right)$ be a group.

$\left({S, \circ}\right)$ is a semigroup by the definition of a group.

By Group has Latin Square Property, the Latin square property holds in $S$.

Sufficient Condition
Let $a \in G$.

By hypothesis:
 * $\exists e \in S: a \circ e = a$

and such an $e$ is unique.

Let $b \in G$.

By hypothesis:
 * $\exists x \in G: x \circ a = b$

and such an $x$ is unique.

Hence:

Also by hypothesis:
 * $\exists a' \in G: a' \circ a = e$

Thus $\left({S, \circ}\right)$ satisfies the right-hand Group Axioms

Thus $\left({S, \circ}\right)$ is a group.