Proof with the help of natural deduction

Question

I have some problems trying to proof the following with the help of natural deduction:

$(S\land \lnot T) \lor (\lnot S \land T),~ \lnot (K \land F),~ T \to K \vdash F \to S$

I don't really know where to start.

Answer 1

Start by proving:

$$S \land \lnot T, \lnot (K \land F), T \to K \vdash F \to S$$

as well as proving

$$\lnot S \land T, \lnot (K \land F), T \to K \vdash F \to S$$

That's proof by cases. Combine the two proofs using $\lor-\text{elim}$ rule.