exact ex_series_scal.
Qed.
End Properties3.

(* Specifications: *)
Lemma Series_scal_l (c : R) (a : nat -> R) :
Series (fun n => c * a n) = c * Series a.
Lemma is_series_scal_r (c : R) (a : nat -> R) (l : R) :
is_series  a l -> is_series  (fun n => (a n) * c) (l * c).
Lemma ex_series_scal_r (c : R) (a : nat -> R) :
ex_series  a -> ex_series  (fun n => a n * c).
Lemma Series_scal_r (c : R) (a : nat -> R) :
Series (fun n => a n * c) = Series a * c.
Lemma is_series_mult_pos (a b : nat -> R) (la lb : R) :
is_series  a la -> is_series  b lb ->
(forall n, 0 <= a n) -> (forall n, 0 <= b n)
-> is_series  (fun n => sum_f_R0 (fun k => a k * b (n - k)%nat) n) (la * lb).
Lemma is_series_mult (a b : nat -> R) (la lb : R) :
is_series  a la -> is_series  b lb
-> ex_series  (fun n => Rabs (a n)) -> ex_series  (fun n => Rabs (b n))
-> is_series  (fun n => sum_f_R0 (fun k => a k * b (n - k)%nat) n) (la * lb).