Fixpoint LNT2LNN_formula (f : Formula) : fol.Formula LNN :=
  match f with
  | fol.equal t1 t2 => fol.equal LNN (LNT2LNN_term t1) (LNT2LNN_term t2)
  | atomic r ts => match r with
                   end
  | fol.impH A B => fol.impH LNN (LNT2LNN_formula A) (LNT2LNN_formula B)
  | fol.notH A => fol.notH LNN (LNT2LNN_formula A)
  | fol.forallH v A => fol.forallH LNN v (LNT2LNN_formula A)
  end.

Lemma translatePrf : forall f, forall axm, Prf LNN axm f -> (forall g, In g axm -> mem _ U g) -> exists Axm : Formulas, (exists prf : Prf LNT Axm (LNN2LNT_formula f), (forall g : Formula, In g Axm -> mem _ V g)) /\ forall v, In v (freeVarListFormula LNT Axm) -> (In v (freeVarListFormula LNN axm)).