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Except one thing, which probably made me forgot the proofs. There are two weird functions in these two proofs: [tex]\phi(x) = x + A^{-1} (y-f(x))[/tex] and [tex]F(x,y) = (f(x,y),y)[/tex].

I can see how they are used and that it really works, but I don't really understand what do the functions really say, if you know what I mean. How did someone figure out that these are the right functions we should use in the proof?