Given information sources $S_1$ and $S_2$, denote by $S_1 \times S_2$ the information source of all pairs of symbols $\left(s_1, s_2\right)$ with $s_i$ a symbol of $S_i(i=1,2)$. Prove that
$$
H\left(S_1 \times S_2\right) \leq \boldsymbol{H}\left(S_1\right)+H\left(S_2\right)
$$
and the equality takes place iff $S_1$ and $S_2$ are independent [i.e., the probability of $\left(s_1, s_2\right)$ is the product of the probability of $s_1$ (in $\left.S_1\right)$ and $s_2$ (in $S_2$ )].