Inverse in Group is Unique

Theorem
Let $\struct {G, \circ}$ be a group.

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

where $e$ is the identity element of $\struct {G, \circ}$.

Also see

 * Inverse in Monoid is Unique