Group has Latin Square Property/Proof 2

Theorem
Let $\left({G, \circ}\right)$ be a Definition:Group.

Let $a, b \in G$.

Then there are unique $x$ and $y$ in $G$ such that:


 * $a \circ x = b$


 * $y \circ a = b$

That is, these equations have unique solutions in $G$.

Proof
We shall prove that this is true for the first equation:

Because the statements:


 * $a \circ x = b$

and


 * $x = a^{-1} \circ b$

are interderivable we may conclude that $x$ is indeed the only solution of the equation.

The proof that the unique solution of $y$ is $b \circ a^{-1}$ in the second equation follows similarly.