Here is an expression that I am trying to reduce and my operations so far:
$$((\lambda x.(x (\lambda z.zy))) (\lambda z.\lambda y. zy) )= (x (\lambda z.zy))[x \to \lambda z.\lambda y. zy ] = ((\lambda z.\lambda y. zy) (\lambda z.zy)) = ((\lambda s.\lambda y. zy) (\lambda z.zy)) = \lambda y. zy$$
Is this procedure correct or am I making some small (or maybe big) mistake?
Thank you very much in advance for your comments.