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(** *    Injection from N to R                            *)
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Fixpoint INR (n:nat) : R :=
  match n with
  | O => 0
  | S O => 1
  | S n => INR n + 1
  end.

Lemma archimed : forall r:R, IZR (up r) > r /\ IZR (up r) - r <= 1.