Inverse in Group is Unique/Proof 2

Proof
Let $\left({G, \circ}\right)$ be a group whose identity element is $e$.

By Group Axioms: $G3$: Inverses, every element of $G$ has at least one inverse.

Suppose that:
 * $\exists b, c \in G: a \circ b = e, a \circ c = e$

that is, that $b$ and $c$ are both inverse elements of $a$.

Then:

So $b = c$ and hence the result.