Section RND_prop.
Open Scope R_scope.
Definition Rnd_DN (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_DN_pt F x (rnd x).
Definition Rnd_UP (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_UP_pt F x (rnd x).
Definition Rnd_ZR (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_ZR_pt F x (rnd x).
Definition Rnd_N (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_N_pt F x (rnd x).
Definition Rnd_NG (F : R -> Prop) (P : R -> R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_NG_pt F P x (rnd x).
Definition Rnd_NA (F : R -> Prop) (rnd : R -> R) :=
forall x : R, Rnd_NA_pt F x (rnd x).

Theorem Rnd_N_pt_abs :
forall F : R -> Prop,
F 0 ->
( forall x, F x -> F (- x) ) ->
forall x f : R,
Rnd_N_pt F x f -> Rnd_N_pt F (Rabs x) (Rabs f).