Inverse of Inverse

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 its inverse is $$x$$. That is:


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

Proof 1
Let $$x \in S$$ be invertible. Then,

$$ $$

So by the definition of $$x^{-1}$$, we see that $$\left({x^{-1}}\right)^{-1} = x$$.

Proof 2
For use when $$G$$ is a group.

Let $$g \in G$$.

$$ $$ $$ $$ $$ $$