Definition A1 (x:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) N.Definition B1 (x:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1))
    N.Definition C1 (x y:R) (N:nat) : R :=
  sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) N.Definition Reste1 (x y:R) (N:nat) : R :=
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
            x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
            y ^ (2 * (N - l))) (pred (N - k))) (pred N).Definition Reste2 (x y:R) (N:nat) : R :=
  sum_f_R0
    (fun k:nat =>
       sum_f_R0
         (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
            x ^ (2 * S (l + k) + 1) *
            ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
            y ^ (2 * (N - l) + 1)) (pred (N - k))) (
    pred N).Definition Reste (x y:R) (N:nat) : R := Reste2 x y N - Reste1 x y (S N).

(* Specifications: *)
Theorem cos_plus_form : forall (x y:R) (n:nat), (0 < n)%nat -> A1 x (S n) * A1 y (S n) - B1 x n * B1 y n + Reste x y n = C1 x y (S n).
Lemma pow_sqr : forall (x:R) (i:nat), x ^ (2 * i) = (x * x) ^ i.
Lemma A1_cvg : forall x:R, Un_cv (A1 x) (cos x).
Lemma C1_cvg : forall x y:R, Un_cv (C1 x y) (cos (x + y)).
Lemma B1_cvg : forall x:R, Un_cv (B1 x) (sin x).