I want to show that
$$ S^{m} \underline{n} = \underline{n+m} $$
where $S$ is successor function and $\underline{m},\underline{n}$ are Church numerals. Note that $S^m (x) = \underbrace{S(S(..(S}_\text{m-times}x)..))$. Also successor function S is defined: $S(\underline{n}) = \underline{n+1}$.
Here's my try:
$$ S^{m} \underline{n} = (\lambda nfx. \underline{m} f (nfx)) \underline{n} = \lambda fx. \underline{m} f (\underline{n} fx) = \lambda fx. f^m (f^n x) = \lambda fx. f^{n+m} x = \underline{n+m} $$
$\square$
Are the steps correct and reasonable? Thanks a bunch!