Question regarding a step in a proof of Zorn's Lemma

Question

I'm trying to understand a proof of Zorn's Lemma. I can follow most of it, but there are a couple of points that I have not been able to comprehend:

1. Why is $z\leq x$?

2. Why is $P(B,y)=P(A,z)$? I can understand that $z$ is also the minimal element of $A-P(A,z)$, but that doesn't seem to get me to the equality.

I would appreciate any help.

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