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.