Let $g$ be a generator element of $G$ and let $x, y \in G$. Since $G$ is cyclic, result
R627: Explicit expression for elements of cyclic group states that there are $n, m \in \mathbb{Z}$ such that $x = g^n$ and $y = g^m$. Now
\begin{equation}
x y = g^n g^m = g^{n + m} = g^{m + n} = g^m g^n = y x
\end{equation}
$\square$
