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(*         *   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.
Set Implicit Arguments.
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(** * Each bounded subset of N has a maximal element *)
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Definition Nbound (I:nat -> Prop) : Prop :=
exists n : nat, (forall i:nat, I i -> (i <= n)%nat).

Lemma Nzorn :
forall I:nat -> Prop,
(exists n : nat, I n) ->
Nbound I -> { n:nat | I n /\ (forall i:nat, I i -> (i <= n)%nat) }.