Have a problem proving K MN=M
By the K combinator definition
$ (\lambda x y.x) M N $
$ ((\lambda x. (\lambda y.x)) M) N $
By the principal axiom of lambda calculus
$ (\lambda y.M) N $
Second application of the principal axiom
$ M[y:= N] $ ?
This would give correct result if M is y-free. Apparently there is an error at one of the steps.