The function after the double-sigma sign can be separated into the product of two terms, the first of which does not depend on $s$ and the second of which does not depend on $r$. Source
Is the derivation below correct? Begin with $$\sum_{p} \sum_{q} f(p)f(q)$$
Move $f(p)$ outside since it is a constant with respect to $q$: $$\sum_{p} \left( f(p) \sum_{q} f(q) \right) $$
Move $\sum_{q} f(q)$ outside since it is a constant with respect to $p$: $$\left( \sum_{q} f(q) \right) \left( \sum_{p}f(p) \right)$$
Yes your derivation is correct.