Introduction to group theory/Uniqueness of Inverses

Proof

Let ${\displaystyle G}$ be a group and let ${\displaystyle g,a,b\in G}$ such that ${\displaystyle a*g=e=g*a}$ and ${\displaystyle b*g=e=g*b}$. Then substituting we obtain ${\displaystyle a*g=b*g}$. By right cancellation ${\displaystyle a=b}$.

Q.E.D.