# Inverse of Inverse/Monoid

## Theorem

Let $\struct {S, \circ}$ be a monoid.

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

Then $x^{-1}$ is also invertible, and:

$\paren {x^{-1} }^{-1} = x$

## Proof

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

$x^{-1}$ is invertible and its inverse is $x$.

That is:

$\paren {x^{-1} }^{-1} = x$

$\blacksquare$