(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Rtrigo_def.
Local Open Scope R_scope.

(***************************************************************)

(***************************************************************)

Definition sin_term (a:R) (i:nat) : R :=
  (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1))).

Definition cos_term (a:R) (i:nat) : R :=
  (-1) ^ i * (a ^ (2 * i) / INR (fact (2 * i))).

Definition sin_approx (a:R) (n:nat) : R := sum_f_R0 (sin_term a) n.

Definition cos_approx (a:R) (n:nat) : R := sum_f_R0 (cos_term a) n.

(**********)
(*
Lemma Alt_PI_4 : Alt_PI <= 4.
Proof.
  assert (H0 := PI_ineq 0).
  elim H0; clear H0; intros _ H0.
  unfold tg_alt, PI_tg in H0; simpl in H0.
  rewrite Rinv_1 in H0; rewrite Rmult_1_r in H0; unfold Rdiv in H0.
  apply Rmult_le_reg_l with (/ 4).
  apply Rinv_0_lt_compat; prove_sup0.
  rewrite <- Rinv_l_sym; [ rewrite Rmult_comm; assumption | discrR ].
Qed.
*)
(**********)
Theorem pre_sin_bound :
  forall (a:R) (n:nat),
    0 <= a ->
    a <= 4 -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)).
forall (a : R) (n : nat),
0 <= a ->
a <= 4 ->
sin_approx a (2 * n + 1) <= sin a <=
sin_approx a (2 * (n + 1))
Proof.
forall (a : R) (n : nat),
0 <= a ->
a <= 4 ->
sin_approx a (2 * n + 1) <= sin a <=
sin_approx a (2 * (n + 1))
  intros; case (Req_dec a 0); intro Hyp_a.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a = 0
sin_approx a (2 * n + 1) <= sin a <=
sin_approx a (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
sin_approx a (2 * n + 1) <= 
sin a <= sin_approx a (2 * (n + 1))
  rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx;
    apply sum_eq_R0 || (symmetry ; apply sum_eq_R0);
      intros; unfold sin_term; rewrite pow_add;
        simpl; unfold Rdiv; rewrite Rmult_0_l;
          ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
sin_approx a (2 * n + 1) <= 
sin a <= sin_approx a (2 * (n + 1))
  unfold sin_approx; cut (0 < a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a ->
sum_f_R0 (sin_term a) (2 * n + 1) <= 
sin a <= sum_f_R0 (sin_term a) (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intro Hyp_a_pos.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (sin_term a) (2 * n + 1) <= 
sin a <= sum_f_R0 (sin_term a) (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite (decomp_sum (sin_term a) (2 * n + 1)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sin_term a 0 +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <= sum_f_R0 (sin_term a) (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite (decomp_sum (sin_term a) (2 * (n + 1))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sin_term a 0 +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
sin_term a 0 +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (sin_term a 0) with a.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  cut
    (sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a - a /\
      sin a - a <= sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1))) ->
      a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a /\
      sin a <= a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1)))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(sum_f_R0 (fun i : nat => sin_term a (S i))
   (Init.Nat.pred (2 * n + 1)) <= 
 sin a - a <=
 sum_f_R0 (fun i : nat => sin_term a (S i))
   (Init.Nat.pred (2 * (n + 1))) ->
 a +
 sum_f_R0 (fun i : nat => sin_term a (S i))
   (Init.Nat.pred (2 * n + 1)) <= 
 sin a <=
 a +
 sum_f_R0 (fun i : nat => sin_term a (S i))
   (Init.Nat.pred (2 * (n + 1)))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intro; apply H1.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  set (Un := fun n:nat => a ^ (2 * S n + 1) / INR (fact (2 * S n + 1))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (pred (2 * n + 1)) with (2 * n)%nat.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n) <=
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (pred (2 * (n + 1))) with (S (2 * n)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n) <=
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (2 * n)) with
  (- sum_f_R0 (tg_alt Un) (2 * n)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (S (2 * n))) with
  (- sum_f_R0 (tg_alt Un) (S (2 * n))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  cut
    (sum_f_R0 (tg_alt Un) (S (2 * n)) <= a - sin a <=
      sum_f_R0 (tg_alt Un) (2 * n) ->
      - sum_f_R0 (tg_alt Un) (2 * n) <= sin a - a <=
      - sum_f_R0 (tg_alt Un) (S (2 * n))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
 a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
 - sum_f_R0 (tg_alt Un) (2 * n) <= 
 sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intro; apply H2.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply alternated_series_ineq.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_decreasing Un
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Un_decreasing, Un; intro;
    cut ((2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1)) ->
a ^ (2 * S (S n0) + 1) / INR (fact (2 * S (S n0) + 1)) <=
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intro; rewrite H3.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ S (S (2 * S n0 + 1)) /
INR (fact (S (S (2 * S n0 + 1)))) <=
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (a ^ S (S (2 * S n0 + 1))) with (a ^ (2 * S n0 + 1) * (a * a)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) /
INR (fact (S (S (2 * S n0 + 1)))) <=
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= a ^ (2 * S n0 + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a * / INR (fact (S (S (2 * S n0 + 1)))) <=
/ INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  left; apply pow_lt; assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a * / INR (fact (S (S (2 * S n0 + 1)))) <=
/ INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n0 + 1))))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 < INR (fact (S (S (2 * S n0 + 1))))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (S (S (2 * S n0 + 1)))) *
(a * a * / INR (fact (S (S (2 * S n0 + 1))))) <=
INR (fact (S (S (2 * S n0 + 1)))) *
/ INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red; intro;
    assert (H5 := eq_sym H4); elim (fact_neq_0 _ H5).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (S (S (2 * S n0 + 1)))) *
(a * a * / INR (fact (S (S (2 * S n0 + 1))))) <=
INR (fact (S (S (2 * S n0 + 1)))) *
/ INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite <- H3; rewrite (Rmult_comm (INR (fact (2 * S (S n0) + 1))));
    rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a * 1 <=
INR (fact (2 * S (S n0) + 1)) *
/ INR (fact (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite Rmult_1_r; rewrite H3; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
    repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a <=
INR (S (S (2 * S n0 + 1))) *
(INR (S (2 * S n0 + 1)) * 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite Rmult_1_r.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a <=
INR (S (S (2 * S n0 + 1))) * INR (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR;
    simpl;
      replace
      (((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1 + 1) *
        ((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1)) with
      (4 * INR n0 * INR n0 + 18 * INR n0 + 20); [ idtac | ring ].
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rle_trans with 20.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rle_trans with 16.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a <= 16
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
16 <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ].
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a * a <= 4²
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
16 <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rsqr_incr_1.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a <= 4
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
16 <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
16 <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
16 <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  now apply IZR_le.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
16 <= 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  now apply IZR_le.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
20 <= 4 * INR n0 * INR n0 + 18 * INR n0 + 20
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite <- (Rplus_0_l 20) at 1;
    apply Rplus_le_compat_r.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4 * INR n0 * INR n0 + 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rplus_le_le_0_compat.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4 * INR n0 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rmult_le_pos.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rmult_le_pos.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 4
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  now apply IZR_le.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply pos_INR.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply pos_INR.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18 * INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rmult_le_pos.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= 18
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  now apply IZR_le.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
0 <= INR n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply pos_INR.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S n0 + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply INR_fact_neq_0.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
INR (fact (2 * S (S n0) + 1)) <> 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply INR_fact_neq_0.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
H3:(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a ^ (2 * S n0 + 1) * (a * a) =
a ^ S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  simpl; ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
n0:nat
(2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv Un 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  assert (H3 := cv_speed_pow_fact a); unfold Un; unfold Un_cv in H3;
    unfold R_dist in H3; unfold Un_cv; unfold R_dist;
      intros; elim (H3 eps H4); intros N H5.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (a ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n0 : nat,
(n0 >= N)%nat ->
Rabs (a ^ n0 / INR (fact n0) - 0) < eps
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs
    (a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)) -
     0) < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  exists N; intros; apply H5.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 + 1 >= N)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (2 * S n0 + 1)%nat with (S (2 * S n0)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(S (2 * S n0) >= N)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold ge; apply le_trans with (2 * S n0)%nat.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(N <= 2 * S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 <= S (2 * S n0))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply le_trans with (2 * S N)%nat.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(N <= 2 * S N)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S N <= 2 * S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 <= S (2 * S n0))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply le_trans with (2 * N)%nat.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(N <= 2 * N)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * N <= 2 * S N)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S N <= 2 * S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 <= S (2 * S n0))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply le_n_2n.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * N <= 2 * S N)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S N <= 2 * S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 <= S (2 * S n0))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S N <= 2 * S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 <= S (2 * S n0))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
(2 * S n0 <= S (2 * S n0))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply le_n_Sn.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
H3:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (a ^ n1 / INR (fact n1) - 0) < eps0
eps:R
H4:eps > 0
N:nat
H5:forall n1 : nat,
(n1 >= N)%nat ->
Rabs (a ^ n1 / INR (fact n1) - 0) < eps
n0:nat
H6:(n0 >= N)%nat
S (2 * S n0) = (2 * S n0 + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - sin a)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold sin.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - (let (a0, _) := exist_sin a² in a * a0))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  destruct (exist_sin (Rsqr a)) as (x,p).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:sin_in a² x
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (a - a * x)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold sin_in, infinite_sum, R_dist in p;
      unfold Un_cv, R_dist;
      intros.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - (a - a * x)) < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  cut (0 < eps / Rabs a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - (a - a * x)) < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intro H4; destruct (p _ H4) as (N,H6).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n0 : nat,
(n0 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
   x) < eps / Rabs a
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n0 - (a - a * x)) < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  exists N; intros.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
Rabs (sum_f_R0 (tg_alt Un) n0 - (a - a * x)) < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (sum_f_R0 (tg_alt Un) n0) with
  (a * (1 - sum_f_R0 (fun i:nat => sin_n i * Rsqr a ^ i) (S n0))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
Rabs
  (a *
   (1 -
    sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) -
   (a - a * x)) < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Rminus; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
    rewrite Ropp_plus_distr; rewrite Ropp_involutive;
      repeat rewrite Rplus_assoc; rewrite (Rplus_comm a);
        rewrite (Rplus_comm (- a)); repeat rewrite Rplus_assoc;
          rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rmult_lt_reg_l with (/ Rabs a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
0 < / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
/ Rabs a *
Rabs
  (a *
   - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0) +
   a * x) < / Rabs a * eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
/ Rabs a *
Rabs
  (a *
   - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0) +
   a * x) < / Rabs a * eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  pattern (/ Rabs a) at 1; rewrite <- (Rabs_Rinv a Hyp_a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
Rabs (/ a) *
Rabs
  (a *
   - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0) +
   a * x) < / Rabs a * eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite <- Rabs_mult, Rmult_plus_distr_l, <- 2!Rmult_assoc, <- Rinv_l_sym;
    [ rewrite Rmult_1_l | assumption ];
      rewrite (Rmult_comm (/ Rabs a)),
        <- Rabs_Ropp, Ropp_plus_distr, Ropp_involutive, Rmult_1_l.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0) +
   - x) < eps * / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
          unfold Rminus, Rdiv in H6.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 +
   - x) < eps * / Rabs a
n0:nat
H5:(n0 >= N)%nat
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0) +
   - x) < eps * / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a apply H6; unfold ge;
            apply le_trans with n0; [ exact H5 | apply le_n_Sn ].
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 - sum_f_R0 (fun i : nat => sin_n i * a² ^ i) (S n0)) =
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite (decomp_sum (fun i:nat => sin_n i * Rsqr a ^ i) (S n0)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 -
 (sin_n 0 * a² ^ 0 +
  sum_f_R0 (fun i : nat => sin_n (S i) * a² ^ S i)
    (Init.Nat.pred (S n0)))) = 
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (sin_n 0) with 1.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
a *
(1 -
 (1 * a² ^ 0 +
  sum_f_R0 (fun i : nat => sin_n (S i) * a² ^ S i)
    (Init.Nat.pred (S n0)))) = 
sum_f_R0 (tg_alt Un) n0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  simpl; rewrite Rmult_1_r; unfold Rminus;
    rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
      rewrite Rplus_0_l; rewrite Ropp_mult_distr_r_reverse;
        rewrite <- Ropp_mult_distr_l_reverse; rewrite scal_sum;
          apply sum_eq.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
forall i : nat,
(i <= n0)%nat ->
sin_n (S i) * (a² * a² ^ i) * - a = tg_alt Un i
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intros; unfold sin_n, Un, tg_alt;
    replace ((-1) ^ S i) with (- (-1) ^ i).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
- (-1) ^ i / INR (fact (2 * S i + 1)) * (a² * a² ^ i) *
- a =
(-1) ^ i *
(a ^ (2 * S i + 1) / INR (fact (2 * S i + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
- (-1) ^ i = (-1) ^ S i
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (a ^ (2 * S i + 1)) with (Rsqr a * Rsqr a ^ i * a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
- (-1) ^ i / INR (fact (2 * S i + 1)) * (a² * a² ^ i) *
- a =
(-1) ^ i *
(a² * a² ^ i * a / INR (fact (2 * S i + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
a² * a² ^ i * a = a ^ (2 * S i + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
- (-1) ^ i = (-1) ^ S i
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Rdiv; ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
a² * a² ^ i * a = a ^ (2 * S i + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
- (-1) ^ i = (-1) ^ S i
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite pow_add; rewrite pow_Rsqr; simpl; ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
       n1 - x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => sin_n i0 * a² ^ i0)
     n1 - x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
i:nat
H7:(i <= n0)%nat
- (-1) ^ i = (-1) ^ S i
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  simpl; ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
1 = sin_n 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold sin_n; unfold Rdiv; simpl; rewrite Rinv_1;
    rewrite Rmult_1_r; reflexivity.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n1 : nat =>
a ^ (2 * S n1 + 1) / INR (fact (2 * S n1 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
     x) < eps0
eps:R
H3:eps > 0
H4:0 < eps / Rabs a
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n1 -
   x) < eps / Rabs a
n0:nat
H5:(n0 >= N)%nat
(0 < S n0)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply lt_O_Sn.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Rdiv; apply Rmult_lt_0_compat.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < eps
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => sin_n i * a² ^ i) n0 -
     x) < eps0
eps:R
H3:eps > 0
0 < / Rabs a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n) ->
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intros; elim H2; intros.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n)
H3:sum_f_R0 (tg_alt Un) (S (2 * n)) <= a - sin a
H4:a - sin a <= sum_f_R0 (tg_alt Un) (2 * n)
- sum_f_R0 (tg_alt Un) (2 * n) <= 
sin a - a <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (sin a - a) with (- (a - sin a)); [ idtac | ring ].
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
H2:sum_f_R0 (tg_alt Un) (S (2 * n)) <= 
a - sin a <= sum_f_R0 (tg_alt Un) (2 * n)
H3:sum_f_R0 (tg_alt Un) (S (2 * n)) <= a - sin a
H4:a - sin a <= sum_f_R0 (tg_alt Un) (2 * n)
- sum_f_R0 (tg_alt Un) (2 * n) <= 
- (a - sin a) <= - sum_f_R0 (tg_alt Un) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  split; apply Ropp_le_contravar; assumption.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with
  (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ].
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (fun i : nat => tg_alt Un i * -1) (S (2 * n)) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply sum_eq; intros; unfold sin_term, Un, tg_alt;
    change ((-1) ^ S i) with (-1 * (-1) ^ i).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
i:nat
H2:(i <= S (2 * n))%nat
(-1) ^ i *
(a ^ (2 * S i + 1) / INR (fact (2 * S i + 1))) * -1 =
-1 * (-1) ^ i *
(a ^ (2 * S i + 1) / INR (fact (2 * S i + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Rdiv; ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (- sum_f_R0 (tg_alt Un) (2 * n)) with
  (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ].
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
sum_f_R0 (fun i : nat => tg_alt Un i * -1) (2 * n) =
sum_f_R0 (fun i : nat => sin_term a (S i)) (2 * n)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply sum_eq; intros.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
i:nat
H2:(i <= 2 * n)%nat
tg_alt Un i * -1 = sin_term a (S i)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold sin_term, Un, tg_alt;
    change ((-1) ^ S i) with (-1 * (-1) ^ i).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i0 : nat => sin_term a (S i0))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
i:nat
H2:(i <= 2 * n)%nat
(-1) ^ i *
(a ^ (2 * S i + 1) / INR (fact (2 * S i + 1))) * -1 =
-1 * (-1) ^ i *
(a ^ (2 * S i + 1) / INR (fact (2 * S i + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold Rdiv; ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (2 * (n + 1))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (2 * (n + 1))%nat with (S (S (2 * n))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = Init.Nat.pred (S (S (2 * n)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (S (2 * n)) = (2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  reflexivity.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (S (2 * n)) = (2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (2 * n + 1)
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (2 * n + 1)%nat with (S (2 * n)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
(2 * n)%nat = Init.Nat.pred (S (2 * n))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = (2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  reflexivity.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
Un:=fun n0 : nat =>
a ^ (2 * S n0 + 1) / INR (fact (2 * S n0 + 1)):nat -> R
S (2 * n) = (2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1))) ->
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  intro; elim H1; intros.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  split.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rplus_le_reg_l with (- a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
- a +
(a +
 sum_f_R0 (fun i : nat => sin_term a (S i))
   (Init.Nat.pred (2 * n + 1))) <= 
- a + sin a
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
    rewrite (Rplus_comm (- a)); apply H2.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
sin a <=
a +
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply Rplus_le_reg_l with (- a).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
H1:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
H2:sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * n + 1)) <= 
sin a - a
H3:sin a - a <=
sum_f_R0 (fun i : nat => sin_term a (S i))
  (Init.Nat.pred (2 * (n + 1)))
- a + sin a <=
- a +
(a +
 sum_f_R0 (fun i : nat => sin_term a (S i))
   (Init.Nat.pred (2 * (n + 1))))
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
    rewrite (Rplus_comm (- a)); apply H3.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
a = sin_term a 0
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  unfold sin_term; simpl; unfold Rdiv; rewrite Rinv_1;
    ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (2 * (n + 1))%nat with (S (S (2 * n))).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < S (S (2 * n)))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
S (S (2 * n)) = (2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply lt_O_Sn.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
S (S (2 * n)) = (2 * (n + 1))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < 2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  replace (2 * n + 1)%nat with (S (2 * n)).
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
(0 < S (2 * n))%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
S (2 * n) = (2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  apply lt_O_Sn.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
Hyp_a_pos:0 < a
S (2 * n) = (2 * n + 1)%nat
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  ring.
a:R
n:nat
H:0 <= a
H0:a <= 4
Hyp_a:a <> 0
0 < a
  inversion H; [ assumption | elim Hyp_a; symmetry ; assumption ].
Qed.

(**********)
Lemma pre_cos_bound :
  forall (a:R) (n:nat),
    - 2 <= a -> a <= 2 ->
    cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)).
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
Proof.
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  cut
    ((forall (a:R) (n:nat),
      0 <= a ->
      a <= 2 ->
      cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))) ->
    forall (a:R) (n:nat),
      - 2 <= a ->
      a <= 2 ->
      cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))).
((forall (a : R) (n : nat),
  0 <= a ->
  a <= 2 ->
  cos_approx a (2 * n + 1) <= cos a <=
  cos_approx a (2 * (n + 1))) ->
 forall (a : R) (n : nat),
 -2 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intros H a n; apply H.
H:(forall (a0 : R) (n0 : nat),
 0 <= a0 ->
 a0 <= 2 ->
 cos_approx a0 (2 * n0 + 1) <= cos a0 <=
 cos_approx a0 (2 * (n0 + 1))) ->
forall (a0 : R) (n0 : nat),
-2 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intros; unfold cos_approx.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (cos_term a0) (2 * n0 + 1) <= cos a0 <=
sum_f_R0 (cos_term a0) (2 * (n0 + 1))
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite (decomp_sum (cos_term a0) (2 * n0 + 1)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
cos_term a0 0 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
sum_f_R0 (cos_term a0) (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite (decomp_sum (cos_term a0) (2 * (n0 + 1))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
cos_term a0 0 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
cos_term a0 0 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (cos_term a0 0) with 1.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  cut
    (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 - 1 /\
      cos a0 - 1 <=
      sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1))) ->
      1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 /\
      cos a0 <=
      1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1)))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(sum_f_R0 (fun i : nat => cos_term a0 (S i))
   (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
 sum_f_R0 (fun i : nat => cos_term a0 (S i))
   (Init.Nat.pred (2 * (n0 + 1))) ->
 1 +
 sum_f_R0 (fun i : nat => cos_term a0 (S i))
   (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
 1 +
 sum_f_R0 (fun i : nat => cos_term a0 (S i))
   (Init.Nat.pred (2 * (n0 + 1)))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intro; apply H2.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  set (Un := fun n:nat => a0 ^ (2 * S n) / INR (fact (2 * S n))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (pred (2 * n0 + 1)) with (2 * n0)%nat.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0) <=
cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (pred (2 * (n0 + 1))) with (S (2 * n0)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0) <=
cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (2 * n0)) with
  (- sum_f_R0 (tg_alt Un) (2 * n0)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (S (2 * n0))) with
  (- sum_f_R0 (tg_alt Un) (S (2 * n0))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  cut
    (sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
      sum_f_R0 (tg_alt Un) (2 * n0) ->
      - sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
      - sum_f_R0 (tg_alt Un) (S (2 * n0))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
 sum_f_R0 (tg_alt Un) (2 * n0) ->
 - sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
 - sum_f_R0 (tg_alt Un) (S (2 * n0))) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intro; apply H3.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply alternated_series_ineq.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_decreasing Un
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold Un_decreasing; intro; unfold Un.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
a0 ^ (2 * S (S n1)) / INR (fact (2 * S (S n1))) <=
a0 ^ (2 * S n1) / INR (fact (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  cut ((2 * S (S n1))%nat = S (S (2 * S n1))).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1)) ->
a0 ^ (2 * S (S n1)) / INR (fact (2 * S (S n1))) <=
a0 ^ (2 * S n1) / INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intro; rewrite H4;
    replace (a0 ^ S (S (2 * S n1))) with (a0 ^ (2 * S n1) * (a0 * a0)).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) /
INR (fact (S (S (2 * S n1)))) <=
a0 ^ (2 * S n1) / INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= a0 ^ (2 * S n1)
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 * / INR (fact (S (S (2 * S n1)))) <=
/ INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply pow_le; assumption.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 * / INR (fact (S (S (2 * S n1)))) <=
/ INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n1))))).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 < INR (fact (S (S (2 * S n1))))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (S (S (2 * S n1)))) *
(a0 * a0 * / INR (fact (S (S (2 * S n1))))) <=
INR (fact (S (S (2 * S n1)))) *
/ INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red; intro;
    assert (H6 := eq_sym H5); elim (fact_neq_0 _ H6).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (S (S (2 * S n1)))) *
(a0 * a0 * / INR (fact (S (S (2 * S n1))))) <=
INR (fact (S (S (2 * S n1)))) *
/ INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite <- H4; rewrite (Rmult_comm (INR (fact (2 * S (S n1)))));
    rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 * 1 <=
INR (fact (2 * S (S n1))) * / INR (fact (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite Rmult_1_r; rewrite H4; do 2 rewrite fact_simpl; do 2 rewrite mult_INR;
    repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 <=
INR (S (S (2 * S n1))) * (INR (S (2 * S n1)) * 1)
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite Rmult_1_r; do 2 rewrite S_INR; rewrite mult_INR; repeat rewrite S_INR;
    simpl;
      replace
      (((0 + 1 + 1) * (INR n1 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n1 + 1) + 1))
    with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ].
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rle_trans with 12.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rle_trans with 4.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 <= 4
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
4 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  change 4 with (Rsqr 2).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 * a0 <= 2²
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
4 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rsqr_incr_1.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 <= 2
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= a0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 2
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
4 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  assumption.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= a0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 2
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
4 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  assumption.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 2
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
4 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  now apply IZR_le.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
4 <= 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  now apply IZR_le.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
12 <= 4 * INR n1 * INR n1 + 14 * INR n1 + 12
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite <- (Rplus_0_l 12) at 1;
    apply Rplus_le_compat_r.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 4 * INR n1 * INR n1 + 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rplus_le_le_0_compat.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 4 * INR n1 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rmult_le_pos.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 4 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rmult_le_pos.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 4
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  now apply IZR_le.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply pos_INR.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply pos_INR.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14 * INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rmult_le_pos.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= 14
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  now apply IZR_le.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
0 <= INR n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply pos_INR.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S n1)) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply INR_fact_neq_0.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
INR (fact (2 * S (S n1))) <> 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply INR_fact_neq_0.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
H4:(2 * S (S n1))%nat = S (S (2 * S n1))
a0 ^ (2 * S n1) * (a0 * a0) = a0 ^ S (S (2 * S n1))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  simpl; ring.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
n1:nat
(2 * S (S n1))%nat = S (S (2 * S n1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv Un 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  assert (H4 := cv_speed_pow_fact a0); unfold Un; unfold Un_cv in H4;
    unfold R_dist in H4; unfold Un_cv; unfold R_dist;
      intros; elim (H4 eps H5); intros N H6; exists N; intros.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
Rabs (a0 ^ (2 * S n1) / INR (fact (2 * S n1)) - 0) <
eps
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply H6; unfold ge; apply le_trans with (2 * S N)%nat.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(N <= 2 * S N)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(2 * S N <= 2 * S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply le_trans with (2 * N)%nat.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(N <= 2 * N)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(2 * N <= 2 * S N)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(2 * S N <= 2 * S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply le_n_2n.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(2 * N <= 2 * S N)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(2 * S N <= 2 * S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps0
eps:R
H5:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs (a0 ^ n2 / INR (fact n2) - 0) < eps
n1:nat
H7:(n1 >= N)%nat
(2 * S N <= 2 * S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - cos a0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold cos.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N)
  (1 - (let (a1, _) := exist_cos a0² in a1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1)) destruct (exist_cos (Rsqr a0)) as (x,p).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:cos_in a0² x
Un_cv (fun N : nat => sum_f_R0 (tg_alt Un) N) (1 - x)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold cos_in, infinite_sum, R_dist in p;
   unfold Un_cv, R_dist; intros.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n1 : nat,
  (n1 >= N)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n1 - x) < eps0
eps:R
H4:eps > 0
exists N : nat,
  forall n1 : nat,
  (n1 >= N)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n1 - (1 - x)) < eps
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  destruct (p _ H4) as (N,H6).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n1 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n1 : nat,
(n1 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n1 -
   x) < eps
exists N0 : nat,
  forall n1 : nat,
  (n1 >= N0)%nat ->
  Rabs (sum_f_R0 (tg_alt Un) n1 - (1 - x)) < eps
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  exists N; intros.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
Rabs (sum_f_R0 (tg_alt Un) n1 - (1 - x)) < eps
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (sum_f_R0 (tg_alt Un) n1) with
  (1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
Rabs
  (1 -
   sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) (S n1) -
   (1 - x)) < eps
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 - sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) (S n1) =
sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold Rminus; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
    repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1);
      rewrite (Rplus_comm (-(1))); repeat rewrite Rplus_assoc;
        rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp;
          rewrite Ropp_plus_distr; rewrite Ropp_involutive;
            unfold Rminus in H6; apply H6.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 +
   - x) < eps
n1:nat
H5:(n1 >= N)%nat
(S n1 >= N)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 - sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) (S n1) =
sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold ge; apply le_trans with n1.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 +
   - x) < eps
n1:nat
H5:(n1 >= N)%nat
(N <= n1)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 +
   - x) < eps
n1:nat
H5:(n1 >= N)%nat
(n1 <= S n1)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 - sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) (S n1) =
sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  exact H5.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 +
   - x) < eps
n1:nat
H5:(n1 >= N)%nat
(n1 <= S n1)%nat
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 - sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) (S n1) =
sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply le_n_Sn.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 - sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) (S n1) =
sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite (decomp_sum (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 -
(cos_n 0 * a0² ^ 0 +
 sum_f_R0 (fun i : nat => cos_n (S i) * a0² ^ S i)
   (Init.Nat.pred (S n1))) = sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (cos_n 0) with 1.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 -
(1 * a0² ^ 0 +
 sum_f_R0 (fun i : nat => cos_n (S i) * a0² ^ S i)
   (Init.Nat.pred (S n1))) = sum_f_R0 (tg_alt Un) n1
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  simpl; rewrite Rmult_1_r; unfold Rminus;
    rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r;
      rewrite Rplus_0_l;
        replace (- sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1)
    with
      (-1 * sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1);
      [ idtac | ring ]; rewrite scal_sum; apply sum_eq;
        intros; unfold cos_n, Un, tg_alt.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
(-1) ^ S i / INR (fact (2 * S i)) * (a0² * a0² ^ i) *
-1 =
(-1) ^ i * (a0 ^ (2 * S i) / INR (fact (2 * S i)))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace ((-1) ^ S i) with (- (-1) ^ i).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
- (-1) ^ i / INR (fact (2 * S i)) * (a0² * a0² ^ i) *
-1 =
(-1) ^ i * (a0 ^ (2 * S i) / INR (fact (2 * S i)))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
- (-1) ^ i = (-1) ^ S i
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (a0 ^ (2 * S i)) with (Rsqr a0 * Rsqr a0 ^ i).
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
- (-1) ^ i / INR (fact (2 * S i)) * (a0² * a0² ^ i) *
-1 = (-1) ^ i * (a0² * a0² ^ i / INR (fact (2 * S i)))
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
a0² * a0² ^ i = a0 ^ (2 * S i)
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
- (-1) ^ i = (-1) ^ S i
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold Rdiv; ring.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
a0² * a0² ^ i = a0 ^ (2 * S i)
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
- (-1) ^ i = (-1) ^ S i
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite pow_Rsqr; reflexivity.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0
       (fun i0 : nat => cos_n i0 * a0² ^ i0) n2 -
     x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i0 : nat => cos_n i0 * a0² ^ i0)
     n2 - x) < eps
n1:nat
H5:(n1 >= N)%nat
i:nat
H7:(i <= n1)%nat
- (-1) ^ i = (-1) ^ S i
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  simpl; ring.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
1 = cos_n 0
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold cos_n; unfold Rdiv; simpl; rewrite Rinv_1;
    rewrite Rmult_1_r; reflexivity.
H:(forall (a1 : R) (n2 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n2 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n2 + 1))) ->
forall (a1 : R) (n2 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n2 + 1) <= cos a1 <=
cos_approx a1 (2 * (n2 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n2 : nat =>
a0 ^ (2 * S n2) / INR (fact (2 * S n2)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
x:R
p:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n2 : nat,
  (n2 >= N0)%nat ->
  Rabs
    (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i)
       n2 - x) < eps0
eps:R
H4:eps > 0
N:nat
H6:forall n2 : nat,
(n2 >= N)%nat ->
Rabs
  (sum_f_R0 (fun i : nat => cos_n i * a0² ^ i) n2 -
   x) < eps
n1:nat
H5:(n1 >= N)%nat
(0 < S n1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply lt_O_Sn.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <=
sum_f_R0 (tg_alt Un) (2 * n0) ->
- sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intros; elim H3; intros; replace (cos a0 - 1) with (- (1 - cos a0));
    [ idtac | ring ].
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
H3:sum_f_R0 (tg_alt Un) (S (2 * n0)) <=
1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0)
H4:sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0
H5:1 - cos a0 <= sum_f_R0 (tg_alt Un) (2 * n0)
- sum_f_R0 (tg_alt Un) (2 * n0) <= - (1 - cos a0) <=
- sum_f_R0 (tg_alt Un) (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  split; apply Ropp_le_contravar; assumption.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with
  (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ].
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
sum_f_R0 (fun i : nat => tg_alt Un i * -1)
  (S (2 * n0)) =
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply sum_eq; intros; unfold cos_term, Un, tg_alt;
    change ((-1) ^ S i) with (-1 * (-1) ^ i).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
i:nat
H3:(i <= S (2 * n0))%nat
(-1) ^ i * (a0 ^ (2 * S i) / INR (fact (2 * S i))) *
-1 =
-1 * (-1) ^ i *
(a0 ^ (2 * S i) / INR (fact (2 * S i)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold Rdiv; ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
- sum_f_R0 (tg_alt Un) (2 * n0) =
sum_f_R0 (fun i : nat => cos_term a0 (S i)) (2 * n0)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with
  (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ];
  apply sum_eq; intros; unfold cos_term, Un, tg_alt;
    change ((-1) ^ S i) with (-1 * (-1) ^ i).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i0 : nat => cos_term a0 (S i0))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
i:nat
H3:(i <= 2 * n0)%nat
(-1) ^ i * (a0 ^ (2 * S i) / INR (fact (2 * S i))) *
-1 =
-1 * (-1) ^ i *
(a0 ^ (2 * S i) / INR (fact (2 * S i)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold Rdiv; ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (2 * (n0 + 1))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = Init.Nat.pred (S (S (2 * n0)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (S (2 * n0)) = (2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  reflexivity.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (S (2 * n0)) = (2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (2 * n0 + 1)
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (2 * n0 + 1)%nat with (S (2 * n0)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
(2 * n0)%nat = Init.Nat.pred (S (2 * n0))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = (2 * n0 + 1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  reflexivity.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
Un:=fun n1 : nat =>
a0 ^ (2 * S n1) / INR (fact (2 * S n1)):nat -> R
S (2 * n0) = (2 * n0 + 1)%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1))) ->
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intro; elim H2; intros; split.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H3:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1
H4:cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H3:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1
H4:cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rplus_le_reg_l with (-(1)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H3:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1
H4:cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
- (1) +
(1 +
 sum_f_R0 (fun i : nat => cos_term a0 (S i))
   (Init.Nat.pred (2 * n0 + 1))) <= - (1) + cos a0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H3:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1
H4:cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
    rewrite (Rplus_comm (-1)); apply H3.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H3:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1
H4:cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
cos a0 <=
1 +
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply Rplus_le_reg_l with (-(1)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
H2:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
H3:sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * n0 + 1)) <= cos a0 - 1
H4:cos a0 - 1 <=
sum_f_R0 (fun i : nat => cos_term a0 (S i))
  (Init.Nat.pred (2 * (n0 + 1)))
- (1) + cos a0 <=
- (1) +
(1 +
 sum_f_R0 (fun i : nat => cos_term a0 (S i))
   (Init.Nat.pred (2 * (n0 + 1))))
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l;
    rewrite (Rplus_comm (-1)); apply H4.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
1 = cos_term a0 0
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  unfold cos_term; simpl; unfold Rdiv; rewrite Rinv_1;
    ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (2 * (n0 + 1))%nat with (S (S (2 * n0))).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < S (S (2 * n0)))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
S (S (2 * n0)) = (2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply lt_O_Sn.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
S (S (2 * n0)) = (2 * (n0 + 1))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  ring.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < 2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  replace (2 * n0 + 1)%nat with (S (2 * n0)).
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
(0 < S (2 * n0))%nat
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
S (2 * n0) = (2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply lt_O_Sn.
H:(forall (a1 : R) (n1 : nat),
 0 <= a1 ->
 a1 <= 2 ->
 cos_approx a1 (2 * n1 + 1) <= cos a1 <=
 cos_approx a1 (2 * (n1 + 1))) ->
forall (a1 : R) (n1 : nat),
-2 <= a1 ->
a1 <= 2 ->
cos_approx a1 (2 * n1 + 1) <= cos a1 <=
cos_approx a1 (2 * (n1 + 1))
a:R
n:nat
a0:R
n0:nat
H0:0 <= a0
H1:a0 <= 2
S (2 * n0) = (2 * n0 + 1)%nat
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  ring.
(forall (a : R) (n : nat),
 0 <= a ->
 a <= 2 ->
 cos_approx a (2 * n + 1) <= cos a <=
 cos_approx a (2 * (n + 1))) ->
forall (a : R) (n : nat),
-2 <= a ->
a <= 2 ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  intros; destruct (total_order_T 0 a) as [[Hlt|Heq]|Hgt].
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hlt:0 < a
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Heq:0 = a
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply H; [ left; assumption | assumption ].
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Heq:0 = a
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  apply H; [ right; assumption | assumption ].
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
  cut (0 < - a).
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a
  intro; cut (forall (x:R) (n:nat), cos_approx x n = cos_approx (- x) n).
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
(forall (x : R) (n0 : nat),
 cos_approx x n0 = cos_approx (- x) n0) ->
cos_approx a (2 * n + 1) <= cos a <=
cos_approx a (2 * (n + 1))
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a
  intro; rewrite H3; rewrite (H3 a (2 * (n + 1))%nat); rewrite cos_sym; apply H.
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
H3:forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
0 <= - a
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
H3:forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
- a <= 2
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a
  left; assumption.
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
H3:forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
- a <= 2
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a
  rewrite <- (Ropp_involutive 2); apply Ropp_le_contravar; exact H0.
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
H2:0 < - a
forall (x : R) (n0 : nat),
cos_approx x n0 = cos_approx (- x) n0
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a
  intros; unfold cos_approx; apply sum_eq; intros;
    unfold cos_term; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg;
      unfold Rdiv; reflexivity.
H:forall (a0 : R) (n0 : nat),
0 <= a0 ->
a0 <= 2 ->
cos_approx a0 (2 * n0 + 1) <= cos a0 <=
cos_approx a0 (2 * (n0 + 1))
a:R
n:nat
H0:-2 <= a
H1:a <= 2
Hgt:0 > a
0 < - a
  apply Ropp_0_gt_lt_contravar; assumption.
Qed.