End Translate_Proof.

Lemma translateProof
: forall (U : fol.System LNN) (V : System),
ClosedSystem LNT V ->
(forall f : fol.Formula LNN,
mem (fol.Formula LNN) U f -> SysPrf V (LNN2LNT_formula f)) ->
forall f : fol.Formula LNN,
folProof.SysPrf LNN U f -> SysPrf V (LNN2LNT_formula f).