End Properties1.

(* Specifications: *)
Lemma Series_incr_1 (a : nat -> R) :
ex_series a -> Series a = a O + Series (fun k => a (S k)).
Lemma Series_incr_n (a : nat -> R) (n : nat) :
(0 < n)%nat -> ex_series a
-> Series a = sum_f_R0 a (pred n)  + Series (fun k => a (n + k)%nat).
Lemma Series_incr_1_aux (a : nat -> R) :
a O = 0 -> Series a = Series (fun k => a (S k)).
Lemma Series_incr_n_aux (a : nat -> R) (n : nat) :
(forall k, (k < n)%nat -> a k = 0)
-> Series a = Series (fun k => a (n + k)%nat).