(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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Local Open Scope R_scope.
(* Boule is French for Ball *)
Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r.
(* General properties of balls. *)

Lemma Boule_convex : forall c d x y z,
Boule c d x -> Boule c d y -> x <= z <= y -> Boule c d z.