Section RInt.
Context {V : CompleteNormedModule R_AbsRing}.
Definition RInt (f : R -> V) (a b : R) := iota (is_RInt f a b).
(** Correction of RInt *)
(* Lemma is_RInt_lim_seq (f : R -> R) (a b : R) : forall If : R, is_RInt f a b If -> is_lim_seq (RInt_val f a b) If. *)
(** ** Usual rewritings *)
(* Lemma RInt_comp_opp (f : R -> V) (a b : R) : ex_RInt f (-a) (-b) -> RInt (fun y => f (- y)) a b = opp (RInt f (-a) (-b)). *)
End RInt.
(** ** Order *)

Lemma RInt_lt (f g : R -> R) (a b : R) :
a < b ->
(forall x : R, a <= x <= b ->continuous g x) ->
(forall x : R, a <= x <= b ->continuous f x) ->
(forall x : R, a < x < b -> f x < g x) ->
RInt f a b < RInt g a b.