End AbelianGroup1.
(** Sum *)
Section Sums.
Context {G : AbelianGroup}.
Definition sum_n_m (a : nat -> G) n m :=
iter_nat plus zero a n m.
Definition sum_n (a : nat -> G) n :=
sum_n_m a O n.

Lemma sum_n_m_plus :
forall (u v : nat -> G) (n m : nat),
sum_n_m (fun k => plus (u k) (v k)) n m = plus (sum_n_m u n m) (sum_n_m v n m).