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 $$g \in G$$.

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