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

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