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_Chasles (a : nat -> G) (n m k : nat) :
(n <= S m)%nat -> (m <= k)%nat
-> sum_n_m a n k = plus (sum_n_m a n m) (sum_n_m a (S m) k).