Group has Latin Square Property

Theorem
For any elements $$a$$ and $$b$$ in a group $$G$$, there exists a unique $$g \in G$$ such that $$ag=b$$. Similarly, there exists a unique $$h \in G$$ such that $$ha=b$$.

Proof
Thus, such a $$g$$ exists.

Suppose $$x \in G$$ where $$ax=b$$. Then,

Thus, $$x$$ is uniquely of the form $$a^{-1}b$$.

To prove the second part of the theorem, let $$h=ba^{-1}$$. The remainder of the proof is similar to the proof above and is therefore omitted.