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The inverse of each element of a group is unique

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