Semigroup is Group Iff Latin Square Property Holds

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 $S$.