Introduction to group theory/Uniqueness of identity proof

Proof:

Let $G$ be a group, and let $e,e^{\prime }\in G$ both be identity elements. Then

$\mathrm{\forall }a\in G,(a*e=a=e*a)$ and $(a*e^{\prime }=a=e^{\prime }*a)$.

Then since $e,e^{\prime }\in G$

$e=e*e^{\prime }=e^{\prime }$ and thus $e=e^{\prime }$.

Q.E.D.