Are inverses of groups unique in general?
The trick here is to pit two inverses $s,t$ of an element $x$ against each other. Let $e$ be the identity element.
By definition, $s\cdot x=e=x\cdot s$ and $t\cdot x=e=x\cdot t$.
We have
$$\begin{align} s&=s\cdot e\\ &=s\cdot (x\cdot t)\\ &=(s\cdot x)\cdot t\\ &=e\cdot t\\ &=t. \end{align}$$