# Introduction to group theory/Uniqueness of identity proof

Let ${\displaystyle G}$ be a group, and let ${\displaystyle e,e'\in G}$ both be identity elements. Then
${\displaystyle \forall a\in G,(a*e=a=e*a)}$ and ${\displaystyle (a*e'=a=e'*a)}$.
Then since ${\displaystyle e,e'\in G}$
${\displaystyle e=e*e'=e'}$ and thus ${\displaystyle e=e'}$.