Inverse in Group is Unique/Proof 2

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

Then every element $x \in G$ has exactly one inverse:
 * $\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x \circ x$

where $e$ is the identity element of $\left({G, \circ}\right)$.

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

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

Then:

So $b = c$ and hence the result.