Inverse of Inverse

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Theorem

General Algebraic Structure

Let $\left({S, \circ}\right)$ be an algebraic structure with an identity element $e$.

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


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


Monoid

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:

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


Group

Let $\left({G, \circ}\right)$ be a group.

Let $g \in G$, with inverse $g^{-1}$.


Then:

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


Ring

Let $\left({R, +, \circ}\right)$ be a ring.

Let $a \in R$ and let $-a$ be the ring negative of $a$.


Then:

$- \left({-a}\right) = a$