Inverse in Group is Unique

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
By the definition of a group, $\left({G, \circ}\right)$ is a monoid each of whose elements has an inverse.

The result follows directly from Inverses in Monoid are Unique.