(** Now, PI is defined *)
Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).

(* Specifications: *)
Lemma Alt_PI_ineq : forall N:nat, sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= Alt_PI / 4 <= sum_f_R0 (tg_alt PI_tg) (2 * N).
Lemma Alt_PI_RGT_0 : 0 < Alt_PI.