Proving that $S_1+S_2 \leq f^{-\omega(1)}$
Let's consider the simpler case in which $f$ only has one input &mdash; the general case is the same, given that you use the correct definitions.

Here is what we want to prove:


  Suppose that $\lim_{n\to\infty} f(n) = \infty$.
  
  If for all $c \geq 0$ there exists $N$ such that for all $n \geq N$, $S_1(n) \leq f(n)^{-c}$ and $S_2(n) \leq f(n)^{-c}$, then for all $c \geq 0$ there exists $N$ such that for all $n \geq N$, $S_1(n) + S_2(n) \leq f(n)^{-c}$.


Let us be given $c \geq 0$. Let $N_1$ be such that $f(n) \geq 2$ for all $n \geq N_1$, and let $N_2$ be such that for all $n \geq N_2$, $S_1(n),S_2(n) \leq f(n)^{-(c+1)}$. Let $N = \max(N_1,N_2)$. Then for $n \geq N$,
$$
S_1(n) + S_2(n) \leq 2f(n)^{-(c+1)} = \frac{2}{f(n)} f(n)^{-c} \leq f(n)^{-c}.
$$