The Renyi entropy of order $\beta$, for a discrete probability distribution $p$ is given by \begin{equation} H_{\beta}(p) = \frac{1}{1 - \beta} ~\log \left( \sum_{i \in S} p(i)^{\beta} \right), \end{equation} where $S$ is the set of all strings in the support of $p$.
As is mentioned here, for two discrete distributions $p$ and $r$ the Renyi entropy of the product distribution $p \times r$ is
\begin{equation} H_{\beta}(p \times r) = H_{\beta}(p) + H_{\beta}(r). \end{equation}
What might be a proof of this fact?