Definition PS_minus (a b : nat -> V) (n : nat) : V := plus (a n) (opp (b n)).

(* Specifications: *)
Lemma is_pseries_minus (a b : nat -> V) (x:K) (la lb : V) : is_pseries a x la -> is_pseries b x lb -> is_pseries (PS_minus a b) x (plus la (opp lb)).
Lemma ex_pseries_minus (a b : nat -> V) (x : K) : ex_pseries a x -> ex_pseries b x -> ex_pseries (PS_minus a b) x.
Lemma PSeries_minus (a b : nat -> R) (x : R) : ex_pseries a x -> ex_pseries b x -> PSeries (PS_minus a b) x = PSeries a x - PSeries b x.
Lemma Abel (a : nat -> R) : Rbar_lt 0 (CV_radius a) -> Rbar_lt (CV_radius a) p_infty -> ex_pseries a (CV_radius a) -> filterlim (PSeries a) (at_left (CV_radius a)) (locally (PSeries a (CV_radius a))).
Lemma PSeries_continuity (a : nat -> R) (x : R) : Rbar_lt (Finite (Rabs x)) (CV_radius a) -> continuity_pt (PSeries a) x.