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(** *        Rmax              *)
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Definition Rmax (x y:R) : R :=
  match Rle_dec x y with
    | left _ => y
    | right _ => x
  end.

Lemma Rmax_case_strong : forall r1 r2 (P:R -> Type), (r2 <= r1 -> P r1) -> (r1 <= r2 -> P r2) -> P (Rmax r1 r2).