Inverse of Inverse/General Algebraic Structure

Theorem

Let $\struct {S, \circ}$ 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 $y$.

Proof

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

Then:

$x \circ y = e = y \circ x$

by definition.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Theorem $4.3$