Inverse of Inverse/General Algebraic Structure
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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 $4.3$