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_pt_refl :
forall (F : R -> Prop) (P : R -> R -> Prop),
forall x, F x -> Rnd_NG_pt F P x x.