Inverse of Inverse

General Algebraic Structures
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $x \in S$ be invertible, and let $y$ be an inverse of $x$.

Then $x$ is also an inverse of $y$.

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

Algebraic Structure
Let $x \in S$ be invertible, where $y$ is an inverse of $x$.

Then:
 * $x \circ y = e = y \circ x$

by definition.

Proof for Monoid
If $\left({S, \circ}\right)$ is a monoid then by definition $\circ$ is associative.

So any inverse of $x$ is unique, and can be denoted $x^{-1}$.

From the result for algebraic structures, $x^{-1}$ is also invertible and its inverse is $x$.

Thus we see that $\left({x^{-1}}\right)^{-1} = x$.

Proof for Group
For use when $G$ is a group.

Let $g \in G$.

Then: