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_sum_n (a:nat -> G) (n m : nat) :
(n <= m)%nat -> sum_n_m a (S n) m = minus (sum_n a m) (sum_n a n).