Inverse of Inverse/Monoid

Theorem
Let $\left({S, \circ}\right)$ be a monoid.

Let $x \in S$ be invertible, and let its inverse be $x^{-1}$.

Then $x^{-1}$ is also invertible, and:
 * $\left({x^{-1}}\right)^{-1} = x$

Proof
By Inverse in Monoid is Unique, any inverse of $x$ is unique, and can be denoted $x^{-1}$.

From Inverse of Inverse in General Algebraic Structure:
 * $x^{-1}$ is invertible and its inverse is $x$.

That is:
 * $\left({x^{-1}}\right)^{-1} = x$