Formal Logic - Natural deduction: Problem with assumptions about exists-negation
It is a bit hard to say exactly which introductions you should perform since we don't have your rule set, but I think the over all strategy should be as follows:
From $\neg (S \lor (P \to Q))$ you get $\neg (P \to Q)$.  Since you have $\neg Q$, you should try to introduce $\neg P$ and thus obtain $P \to Q$.  That should be a contradiction and you can conclude $P$.