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 Inf_fct_min (f : R -> R) (a b : R) (m : R) :
(forall x, Rmin a b < x < Rmax a b -> m <= f x) ->
is_finite (Inf_fct f a b).