Definition Rnd_NG_pt_unicity_prop F P :=
  forall x d u,
  Rnd_DN_pt F x d -> Rnd_N_pt F x d ->
  Rnd_UP_pt F x u -> Rnd_N_pt F x u ->
  P x d -> P x u -> d = u.

Theorem Rnd_NG_unicity : forall (F : R -> Prop) (P : R -> R -> Prop), Rnd_NG_pt_unicity_prop F P -> forall rnd1 rnd2 : R -> R, Rnd_NG F P rnd1 -> Rnd_NG F P rnd2 -> forall x, rnd1 x = rnd2 x.