Introduction to group theory/Uniqueness of identity proof


 * Proof:

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

Then since $$e,e' \in G$$
 * $$e=e*e'=e'$$ and thus $$e=e'$$.
 * Q.E.D.