Chapter 3.2, Problem 3E

Prove part $e$ of Theorem $3.4$.

Theorem 3.4: Properties of Group Elements

Let $G$ be a group with respect to a binary operation that is written as multiplication.

The identity element $e$ in $G$ is unique.

For each $x\in G$, the inverse $x^{-1}$ in $G$ is unique.

For each $x\in G,{(x^{-1})}^{-1}=x$.

Reverse order law: For any $x$ and $y$ in $G$, ${(xy)}^{-1}=y^{-1}{x}^{-1}$.

Cancellation laws: If $a,x,$ and $y$ are in $G$, then either of the equations $ax=ay$ or $xa=ya$ implies that $x=y$.