Composition of tempered distribution with Schwartz function

Question

Let $h\in S(R^{n+1})$ and $f\in S^\prime(R)$. Is it true that the map $R^n\ni(s_1,...,s_n)\mapsto f(h(\cdot,s_1,...,s_n)) $ is in $S(R^n) $?

I think it is true (for example, if $h $ has separated variables). But, how to prove it in general?