Inverse of Inverse of Subset of Group

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

Let $X \subseteq G$.

Then $\left({ X^{-1} }\right)^{-1} = X$.

That is, the inverse of the inverse of $X$ is $X$ itself.

Proof
By the definition of inverse subset:


 * $X^{-1} = \left\{{ x^{-1} : x \in X }\right\}$

Thus: