See group

Suppose $b,b^{\prime }$ are inverses of $a$. Then $b=b^{\prime }$.

Proof

By definition of inverse,
$ab=e=ba$
$a{b}^{\prime }=e=b^{\prime }a$

$ab=a{b}^{\prime }$
$bab=ba{b}^{\prime }$
$(ba)b=(ba)b^{\prime }$
$eb=e{b}^{\prime }$
$b=b^{\prime }$

QED