Semigroup is Group Iff Latin Square Property Holds/Proof 2

Necessary Condition
Let $\struct {S, \circ}$ be a group.

$\struct {S, \circ}$ 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 $a \in G$.

We have :
 * $\exists e \in S: a \circ e = a$

and such an $e$ is unique.

Let $b \in G$.

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

and such an $x$ is unique.

Hence:

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

Thus $\struct {S, \circ}$ satisfies the right-hand Group Axioms

Thus $\struct {S, \circ}$ is a group.