(********************************************************)
(** *            Proof of Heine's theorem               *)
(********************************************************)
Definition uniform_continuity (f:R -> R) (X:R -> Prop) : Prop :=
forall eps:posreal,
exists delta : posreal,
(forall x y:R,
X x -> X y -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps).

Lemma domain_P1 :
forall X:R -> Prop,
~ (exists y : R, X y) \/
(exists y : R, X y /\ (forall x:R, X x -> x = y)) \/
(exists x : R, (exists y : R, X x /\ X y /\ x <> y)).