The inverse of each element of a group is unique

Suppose two elements $y$ and $z$ are both inverses of the element $x$. Then $$\begin{split} y & = ey & \qquad\text{where e is the identity of the group} \\ & = (zx)y & \qquad\text{since z is an inverse for x} \\ & = z(xy) & \qquad\text{because the multiplication is associative} \\ & = ze & \qquad\text{since y is an inverse for x too} \\ & = z & \qquad\text{as e is the identity} \end{split}$$