(** * Operations *)
(** Additive operators *)
Section Properties2.
Context {K : AbsRing} {V : NormedModule K}.

(* Specifications: *)
Lemma is_series_opp (a : nat -> V) (la : V) :
is_series a la
-> is_series (fun n => opp (a n)) (opp la).
Lemma ex_series_opp (a : nat -> V) :
ex_series a
-> ex_series (fun n => opp (a n)).
Lemma Series_opp (a : nat -> R) :
Series (fun n => - a n) = - Series a.
Lemma is_series_plus (a b : nat -> V) (la lb : V) :
is_series a la -> is_series b lb
-> is_series (fun n => plus (a n)  (b n)) (plus la  lb).
Lemma ex_series_plus (a b : nat -> V) :
ex_series a -> ex_series b
-> ex_series (fun n => plus (a n) (b n)).
Lemma is_series_minus (a b : nat -> V) (la lb : V) :
is_series a la -> is_series b lb
-> is_series (fun n => plus (a n) (opp (b n))) (plus la (opp lb)).
Lemma ex_series_minus (a b : nat -> V) :
ex_series a -> ex_series b
-> ex_series (fun n => plus (a n) (opp (b n))).