Entropy of composition

Question

Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (countable measurable) partition of $X$. Show that $h(ST,A)≤h(S,A)+h(T,A)$. If $S=T^k$ (for a natural $k$), it's rather easy. I couldnt get any further. Any help/reference will be appreciated. Thanks.