Any help would really be greatly appreciated. I've attached my workings but I'm not too sure if what I've done is right or not. Thank you ever so much in advanced!
Show that the set $S=\{s_1,s_2\}$ is inconsistent by forming the sentences $s_i$ as premises and deriving a contradiction. $$S =\{ \forall x (Px \rightarrow Qx)\rightarrow\forall x (Px\rightarrow \neg Rx) , \exists x(Px \land Rx)\land\forall x(Px\rightarrow Qx)\}$$
My ideas:
Your proof is largely correct, but there are a number of minor things that need to be corrected, and in which the proof can be shortened quite a bit:
Lines 3 and 4: I am not sure about your rule, which looks like $\top I$ ... This should be $\land E$ or Simplification
Line 7: as we will see, there is no need to start a subproof here. And you already have this line on line 4. So this line can be removed
Line 8 can be removed as well
Line 9: again, no subproof necessary. Instead, get this statement from 1 and 4 through $\rightarrow E$, or Modus Ponens.
Line 11: first you need to isolate $Pa$ from 5 before you can get $\neg Ra$.
Line 12: again, first isolate $Ra$ by Simplication or $\land E$ from 5 before conjuncting $Ra$ and $\neg Ra$
... And line12 is your contradiction, hence your last line. So no more lines needed after this ... Which is why you don't need any of those subproofs.