I have some trouble understanding why this proof is a proof:

Lemma:

Proof:

Ok, this is just conditional independence, x independent of y given z.

This is rewritten according to the rule of conditional independence P(A|B) = P(A & B) / P(B).

Bottom up the same is done for the right hand side of the lemma, but I don't get how the following and previous step are equal.. Why does this prove anything?