Fixpoint freeVarListFormula (l : fol.Formulas L) : 
 list nat :=
  match l with
  | nil => nil (A:=nat)
  | f :: l => freeVarFormula f ++ freeVarListFormula l
  end.

Lemma In_freeVarListFormula : forall (v : nat) (f : fol.Formula L) (F : fol.Formulas L), In v (freeVarFormula f) -> In f F -> In v (freeVarListFormula F).