(** ** Norm *)
Section norm_RInt.
Context {V : NormedModule R_AbsRing}.

(* Specifications: *)
Lemma norm_RInt_le :
forall (f : R -> V) (g : R -> R) (a b : R) (lf : V) (lg : R),
a <= b ->
(forall x, a <= x <= b -> norm (f x) <= g x) ->
is_RInt f a b lf ->
is_RInt g a b lg ->
norm lf <= lg.
Lemma norm_RInt_le_const :
forall (f : R -> V) (a b : R) (lf : V) (M : R),
a <= b ->
(forall x, a <= x <= b -> norm (f x) <= M) ->
is_RInt f a b lf ->
norm lf <= (b - a) * M.
Lemma norm_RInt_le_const_abs :
forall (f : R -> V) (a b : R) (lf : V) (M : R),
(forall x, Rmin a b <= x <= Rmax a b -> norm (f x) <= M) ->
is_RInt f a b lf ->
norm lf <= Rabs (b - a) * M.