Proof of A Binary Operation Has at Most One Neutral Element

By Associative and Commutative Binary Operations, and Neutral Elements §neutral, the neutrality of $e'$ gives $x\ast e'=x$ for every $x\in X$, in particular $e\ast e'=e$; and the neutrality of $e$ gives $e\ast x=x$ for every $x\in X$, in particular $e\ast e'=e'$. Hence

$$e=e\ast e'=e'.$$