(related to Proposition: Uniqueness of Inverse Elements)
- By hypothesis, $x\in X$ has at least one inverse element $y\in X.$
- Let $z\in X$ be another inverse element of $x.$
- By definition of inverse elements, we have $x\ast y=y\ast x=e$ and $x\ast z=z\ast x=e,$ where $e$ is the unique neutral element.
- Since "$\ast$" is an associative, it follows $z=z\ast e=z\ast(x\ast y)=(z\ast x)\ast y=e\ast y=y.$
- Therefore, $y$ is unique.
- Fischer, Gerd: "Lehrbuch der Algebra", Springer Spektrum, 2017, 4th Edition