∀x.(P(x) ∨ Q(x)) ⊢ ∀x.P(x) ∨ ∃x.Q(x): How to prove the validity of this sequent in predicate logic?

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How to prove the validity of this sequent in predicate logic?

$\forall x (P(x) \lor Q(x)) ⊢ \forall x P(x) \lor \exists xQ(x)$:

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Please see the following image.

It was a difficult one! A lot of things to do... :-/