First of all, result R810: Group centre is nonempty shows that $Z(G)$ is not empty. Therefore, if $g \in Z(G)$, then $g x g^{-1} = x$ and thus $g x = x g$ for every $x \in G$. Particularly, $g x = x g$ for every $x \in Z(G)$, which proves the claim. $\square$