Section Translate_Proof.
Variable U : fol.System LNN.
Variable V : System.
Hypothesis
AxiomsOK :
forall f : fol.Formula LNN,
mem _ U f ->
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 (freeVarFormula LNN f)).

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)).