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(** *            Proof of Heine's theorem               *)
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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 is_lub_u : forall (E:R -> Prop) (x y:R), is_lub E x -> is_lub E y -> x = y.