Definition Sup_fct (f : R -> R) (a b : R) : Rbar :=
  match Req_EM_T a b with
    | right Hab => Lub_Rbar (fun y => exists x, y = f x /\ Rmin a b < x < Rmax a b)
    | left _ => Finite 0
  end.Definition Inf_fct (f : R -> R) (a b : R) : Rbar :=
  match Req_EM_T a b with
    | right Hab => Glb_Rbar (fun y => exists x, y = f x /\ Rmin a b < x < Rmax a b)
    | left _ => Finite 0
  end.

Lemma Sup_fct_bound (f : R -> R) (a b : R) : Sup_fct f a b = Sup_fct f b a.