An additional question about the paternal function in Lacan's sexuation formulae

Our friend who had posed me a question about Lacan's formulae of sexuation posed me another question : why is the paternal function ($x) ØΦ(x) situated on the place of $, not on the place of S1 ?

As far as the male sexuation formulae are concerned, the exact place of the paternal function ($x) ØΦ(x) is indifferent because it is only supposed that there ex-sists an x such as ØΦ(x). What is true is Ø($x) ØΦ(x) : there doesn't ex-sist any x which has the impossible phallus of the impossible sexual relation, as it is formulated in the female sexuation formulae.

When Lacan says in his Séminaire XVII L'envers de la psychanalyse that the Father is castrated and dead since origin, that Father is the empty locality as such of the 
ex-sistence (coloured red).

So you can say, if you like, that in the male sexuation formulae the ex-sistent Father is S1 in the discourse of university where S1 can be considered as the empty hole of the inexistent Urvater killed by his sons.

What matters is that the sons (i.e. all the men) are believing in the ex-sistence of the dead Father who would castrate them so that they can not be free from castration anxiety.