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_unicity :
forall (F : R -> Prop) (P : R -> R -> Prop),
Rnd_NG_pt_unicity_prop F P ->
forall x f1 f2 : R,
Rnd_NG_pt F P x f1 -> Rnd_NG_pt F P x f2 ->
f1 = f2.