Inverse of Inverse/Monoid

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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:

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

$\blacksquare$


Sources