1. A -> X
2. (A -> Y) -> (K v X)
3. (A & X) -> Y
4. ~K / X
This one is hard.
Again by indirect argument.
1. A -> X
2. (A -> Y) -> (K v X)
3. (A & X) -> Y
4. ~K / X
5. Assume ~X
6. ~A MT[1]
7. (~K^~X)->~(A->Y) transposition[2]
8. (~K^~X)->(A^~Y) negation[7]
9. (~K^~X) conjunction [4,5]
10 (A^~Y) MP[8,9]
11 A simplification[10]
12 (A^~A) conj. [11,6]
There is the contradiction.