Definition PS_decr_1 (a : nat -> V) (n : nat) : V := a (S n).

(* Specifications: *)
Lemma is_pseries_decr_1 (a : nat -> V) (x y : K) (l : V) :
mult y x = one -> is_pseries a x l
-> is_pseries (PS_decr_1 a) x (scal y (plus l (opp (a O)))).
Lemma ex_pseries_decr_1 (a : nat -> V) (x : K) :
(x = zero \/ exists y, mult y x = one) ->
ex_pseries a x -> ex_pseries (PS_decr_1 a) x.