(************************************************************************)
(*         *   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.
Require Import Cos_rel.
Require Import Max.
Require Import Omega.
Local Open Scope nat_scope.
Local Open Scope R_scope.

Definition Majxy (x y:R) (n:nat) : R :=
  Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n).

Lemma Majxy_cv_R0 : forall x y:R, Un_cv (Majxy x y) 0.
forall x y : R, Un_cv (Majxy x y) 0
Proof.
forall x y : R, Un_cv (Majxy x y) 0
  intros.
x, y:R
Un_cv (Majxy x y) 0
  set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Un_cv (Majxy x y) 0
  set (C0 := C ^ 4).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
Un_cv (Majxy x y) 0
  cut (0 < C).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C -> Un_cv (Majxy x y) 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  intro.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
Un_cv (Majxy x y) 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  cut (0 < C0).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0 -> Un_cv (Majxy x y) 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  intro.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
Un_cv (Majxy x y) 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  assert (H1 := cv_speed_pow_fact C0).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:Un_cv (fun n : nat => C0 ^ n / INR (fact n)) 0
Un_cv (Majxy x y) 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold Un_cv in H1; unfold R_dist in H1.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C0 ^ n / INR (fact n) - 0) < eps
Un_cv (Majxy x y) 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold Un_cv; unfold R_dist; intros.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C0 ^ n / INR (fact n) - 0) < eps0
eps:R
H2:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  cut (0 < eps / C0);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
	[ assumption | apply Rinv_0_lt_compat; assumption ] ].
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C0 ^ n / INR (fact n) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  elim (H1 (eps / C0) H3); intros N0 H4.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C0 ^ n / INR (fact n) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n : nat,
(n >= N0)%nat ->
Rabs (C0 ^ n / INR (fact n) - 0) < eps / C0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  exists N0; intros.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (Majxy x y n - 0) < eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  replace (Majxy x y n) with (C0 ^ S n / INR (fact n)).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (C0 ^ S n / INR (fact n) - 0) < eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  simpl.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (C0 * C0 ^ n / INR (fact n) - 0) < eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  apply Rmult_lt_reg_l with (Rabs (/ C0)).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
0 < Rabs (/ C0)
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0) * Rabs (C0 * C0 ^ n / INR (fact n) - 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  apply Rabs_pos_lt.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
/ C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0) * Rabs (C0 * C0 ^ n / INR (fact n) - 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  apply Rinv_neq_0_compat.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0) * Rabs (C0 * C0 ^ n / INR (fact n) - 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  red; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0) * Rabs (C0 * C0 ^ n / INR (fact n) - 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite <- Rabs_mult.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0 * (C0 * C0 ^ n / INR (fact n) - 0)) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold Rminus; rewrite Rmult_plus_distr_l.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs
  (/ C0 * (C0 * C0 ^ n / INR (fact n)) + / C0 * - 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite Ropp_0; rewrite Rmult_0_r.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0 * (C0 * C0 ^ n / INR (fact n)) + 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold Rdiv; repeat rewrite <- Rmult_assoc.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (/ C0 * C0 * C0 ^ n * / INR (fact n) + 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite <- Rinv_l_sym.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (1 * C0 ^ n * / INR (fact n) + 0) <
Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite Rmult_1_l.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (C0 ^ n * / INR (fact n) + 0) < Rabs (/ C0) * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite (Rabs_right (/ C0)).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (C0 ^ n * / INR (fact n) + 0) < / C0 * eps
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
/ C0 >= 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite <- (Rmult_comm eps).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (C0 ^ n * / INR (fact n) + 0) < eps * / C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
/ C0 >= 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  replace (C0 ^ n * / INR (fact n) + 0) with (C0 ^ n * / INR (fact n) - 0);
    [ idtac | ring ].
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
Rabs (C0 ^ n * / INR (fact n) - 0) < eps * / C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
/ C0 >= 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold Rdiv in H4; apply H4; assumption.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
/ C0 >= 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  apply Rle_ge; left; apply Rinv_0_lt_compat; assumption.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 <> 0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  red; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0).
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) = Majxy x y n
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold Majxy.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
C0 ^ S n / INR (fact n) =
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) /
INR (fact n)
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold C0.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
(C ^ 4) ^ S n / INR (fact n) =
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) /
INR (fact n)
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  rewrite pow_mult.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
H0:0 < C0
H1:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps0
eps:R
H2:eps > 0
H3:0 < eps / C0
N0:nat
H4:forall n0 : nat,
(n0 >= N0)%nat ->
Rabs (C0 ^ n0 / INR (fact n0) - 0) < eps / C0
n:nat
H5:(n >= N0)%nat
(C ^ 4) ^ S n / INR (fact n) =
(Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ 4) ^ S n /
INR (fact n)
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold C; reflexivity.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
H:0 < C
0 < C0
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  unfold C0; apply pow_lt; assumption.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < C
  apply Rlt_le_trans with 1.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
0 < 1
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
1 <= C
  apply Rlt_0_1.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
1 <= C
  unfold C.
x, y:R
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C0:=C ^ 4:R
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
  apply RmaxLess1.
Qed.

Lemma reste1_maj :
  forall (x y:R) (N:nat),
    (0 < N)%nat -> Rabs (Reste1 x y N) <= Majxy x y (pred N).
forall (x y : R) (N : nat),
(0 < N)%nat ->
Rabs (Reste1 x y N) <= Majxy x y (Init.Nat.pred N)
Proof.
forall (x y : R) (N : nat),
(0 < N)%nat ->
Rabs (Reste1 x y N) <= Majxy x y (Init.Nat.pred N)
  intros.
x, y:R
N:nat
H:(0 < N)%nat
Rabs (Reste1 x y N) <= Majxy x y (Init.Nat.pred N)
  set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs (Reste1 x y N) <= Majxy x y (Init.Nat.pred N)
  unfold Reste1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs
  (sum_f_R0
     (fun k : nat =>
      sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l))) (Init.Nat.pred (N - k)))
     (Init.Nat.pred N)) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	Rabs
        (sum_f_R0
          (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
            x ^ (2 * S (l + k)) *
            ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
            y ^ (2 * (N - l))) (pred (N - k)))) (
	      pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs
  (sum_f_R0
     (fun k : nat =>
      sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l))) (Init.Nat.pred (N - k)))
     (Init.Nat.pred N)) <=
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l))) (Init.Nat.pred (N - k))))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l))) (Init.Nat.pred (N - k))))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply
    (Rsum_abs
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
          x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
          y ^ (2 * (N - l))) (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l))) (Init.Nat.pred (N - k))))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          Rabs
          ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
            x ^ (2 * S (l + k)) *
            ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
            y ^ (2 * (N - l)))) (pred (N - k))) (
	      pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l))) (Init.Nat.pred (N - k))))
  (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply sum_Rle.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
forall n : nat,
(n <= Init.Nat.pred N)%nat ->
Rabs
  (sum_f_R0
     (fun l : nat =>
      (-1) ^ S (l + n) / INR (fact (2 * S (l + n))) *
      x ^ (2 * S (l + n)) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
      y ^ (2 * (N - l))) (Init.Nat.pred (N - n))) <=
sum_f_R0
  (fun l : nat =>
   Rabs
     ((-1) ^ S (l + n) / INR (fact (2 * S (l + n))) *
      x ^ (2 * S (l + n)) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
      y ^ (2 * (N - l)))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
Rabs
  (sum_f_R0
     (fun l : nat =>
      (-1) ^ S (l + n) / INR (fact (2 * S (l + n))) *
      x ^ (2 * S (l + n)) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
      y ^ (2 * (N - l))) (Init.Nat.pred (N - n))) <=
sum_f_R0
  (fun l : nat =>
   Rabs
     ((-1) ^ S (l + n) / INR (fact (2 * S (l + n))) *
      x ^ (2 * S (l + n)) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
      y ^ (2 * (N - l)))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply
    (Rsum_abs
      (fun l:nat =>
	(-1) ^ S (l + n) / INR (fact (2 * S (l + n))) * x ^ (2 * S (l + n)) *
	((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
	y ^ (2 * (N - l))) (pred (N - n))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          / INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
          C ^ (2 * S (N + k))) (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) *
         x ^ (2 * S (l + k)) *
         ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
         y ^ (2 * (N - l)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   Rabs
     ((-1) ^ S (l + n) / INR (fact (2 * S (l + n))) *
      x ^ (2 * S (l + n)) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) *
      y ^ (2 * (N - l)))) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun l : nat =>
   / INR (fact (2 * S (l + n)) * fact (2 * (N - l))) *
   C ^ (2 * S (N + n))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs
  ((-1) ^ S (n0 + n) / INR (fact (2 * S (n0 + n))) *
   x ^ (2 * S (n0 + n)) *
   ((-1) ^ (N - n0) / INR (fact (2 * (N - n0)))) *
   y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold Rdiv; repeat rewrite Rabs_mult.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs ((-1) ^ S (n0 + n)) *
Rabs (/ INR (fact (2 * S (n0 + n)))) *
Rabs (x ^ (2 * S (n0 + n))) *
(Rabs ((-1) ^ (N - n0)) *
 Rabs (/ INR (fact (2 * (N - n0))))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  do 2 rewrite pow_1_abs.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 * Rabs (/ INR (fact (2 * S (n0 + n)))) *
Rabs (x ^ (2 * S (n0 + n))) *
(1 * Rabs (/ INR (fact (2 * (N - n0))))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  do 2 rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (/ INR (fact (2 * S (n0 + n)))) *
Rabs (x ^ (2 * S (n0 + n))) *
Rabs (/ INR (fact (2 * (N - n0)))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) *
Rabs (x ^ (2 * S (n0 + n))) *
Rabs (/ INR (fact (2 * (N - n0)))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite (Rabs_right (/ INR (fact (2 * (N - n0))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) *
Rabs (x ^ (2 * S (n0 + n))) *
/ INR (fact (2 * (N - n0))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) *
Rabs (x ^ (2 * S (n0 + n))) *
/ INR (fact (2 * (N - n0))) *
Rabs (y ^ (2 * (N - n0))) <=
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * (N - n0)))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) *
Rabs (x ^ (2 * S (n0 + n))) *
/ INR (fact (2 * (N - n0))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * S (n0 + n))) *
/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) *
(Rabs (x ^ (2 * S (n0 + n))) *
 (/ INR (fact (2 * (N - n0))) *
  Rabs (y ^ (2 * (N - n0))))) <=
/ INR (fact (2 * S (n0 + n))) *
(/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= / INR (fact (2 * S (n0 + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n))) *
(/ INR (fact (2 * (N - n0))) *
 Rabs (y ^ (2 * (N - n0)))) <=
/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n))) *
(/ INR (fact (2 * (N - n0))) *
 Rabs (y ^ (2 * (N - n0)))) <=
/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n))) *
/ INR (fact (2 * (N - n0))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) *
Rabs (x ^ (2 * S (n0 + n))) *
Rabs (y ^ (2 * (N - n0))) <=
/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) *
(Rabs (x ^ (2 * S (n0 + n))) *
 Rabs (y ^ (2 * (N - n0)))) <=
/ INR (fact (2 * (N - n0))) * C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= / INR (fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n))) *
Rabs (y ^ (2 * (N - n0))) <= C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n))) *
Rabs (y ^ (2 * (N - n0))) <= C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  do 2 rewrite <- RPow_abs.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * Rabs y ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with (Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * Rabs y ^ (2 * (N - n0)) <=
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs x ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y ^ (2 * (N - n0)) <= C ^ (2 * (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply pow_le; apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y ^ (2 * (N - n0)) <= C ^ (2 * (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply pow_incr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs y <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  split.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs y
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with (C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * (N - n0)) * Rabs x ^ (2 * S (n0 + n)) <=
C ^ (2 * (N - n0)) * C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C ^ (2 * (N - n0))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) <= C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) <= C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) <= C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  left; apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) <= C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n)) <= C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply pow_incr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs x <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  split.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs x
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold C; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x <= Rmax (Rabs x) (Rabs y)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rmax (Rabs x) (Rabs y) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rmax (Rabs x) (Rabs y) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) <=
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  right.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) =
C ^ (2 * S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  replace (2 * S (N + n))%nat with (2 * (N - n0) + 2 * S (n0 + n))%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) =
C ^ (2 * (N - n0) + 2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 2 * S (n0 + n))%nat =
(2 * S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite pow_add.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0)) =
C ^ (2 * (N - n0)) * C ^ (2 * S (n0 + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 2 * S (n0 + n))%nat =
(2 * S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 2 * S (n0 + n))%nat =
(2 * S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_eq; rewrite plus_INR; do 3 rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR 2 * INR (N - n0) + INR 2 * INR (S (n0 + n)) =
INR 2 * INR (S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite minus_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR 2 * (INR N - INR n0) + INR 2 * INR (S (n0 + n)) =
INR 2 * INR (S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n0 <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  repeat rewrite S_INR; do 2 rewrite plus_INR; ring.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n0 <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_trans with (pred (N - n)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n0 <= Init.Nat.pred (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred (N - n) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  exact H1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred (N - n) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_S_n.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(S (Init.Nat.pred (N - n)) <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  replace (S (pred (N - n))) with (N - n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_trans with N.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply (fun p n m:nat => plus_le_reg_l n m p) with n.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n + (N - n) <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= n + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_plus_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N <= S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply S_pred with 0%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(0 < N - n)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply plus_lt_reg_l with n.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n + 0 < n + (N - n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- le_plus_minus.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n + 0 < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  replace (n + 0)%nat with n; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_lt_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N < N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply lt_pred_n_n; assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n))) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          / INR (fact (2 * S (l + k)) * fact (2 * (N - l))) * C ^ (4 * N))
        (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (2 * S (N + k))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   / INR (fact (2 * S (l + n)) * fact (2 * (N - l))) *
   C ^ (2 * S (N + n))) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun l : nat =>
   / INR (fact (2 * S (l + n)) * fact (2 * (N - l))) *
   C ^ (4 * N)) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (2 * S (N + n)) <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <=
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (N + n)) <= C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 < INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (N + n)) <= C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (N + n)) <= C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_pow.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (N + n) <= 4 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (N + n) <= 4 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (N + n) <= 2 * (2 * N))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply (fun m n p:nat => mult_le_compat_l p n m).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(S (N + n) <= 2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  replace (2 * N)%nat with (S (N + pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(S (N + n) <= S (N + Init.Nat.pred N))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + Init.Nat.pred N) = (2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_n_S.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N + n <= N + Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + Init.Nat.pred N) = (2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply plus_le_compat_l; assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + Init.Nat.pred N) = (2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite pred_of_minus.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + (N - 1)) = (2 * N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0 (fun l:nat => C ^ (4 * N) * Rsqr (/ INR (fact (S (N + k)))))
        (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR (fact (2 * S (l + k)) * fact (2 * (N - l))) *
      C ^ (4 * N)) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   / INR (fact (2 * S (l + n)) * fact (2 * (N - l))) *
   C ^ (4 * N)) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun _ : nat =>
   C ^ (4 * N) * (/ INR (fact (S (N + n))))²)
  (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) *
C ^ (4 * N) <=
C ^ (4 * N) * (/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite <- (Rmult_comm (C ^ (4 * N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (4 * N) *
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) <=
C ^ (4 * N) * (/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  left; apply Rlt_le_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 < 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with
    (Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) <=
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Rdiv;
    do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) <=
/ INR (fact (2 * S (N + n))) *
Binomial.C (2 * S (N + n)) (S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= / INR (fact (2 * S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) <=
Binomial.C (2 * S (N + n)) (S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) <=
Binomial.C (2 * S (N + n)) (S (N + n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply C_maj.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (n0 + n) <= 2 * S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) <=
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  right.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (S (N + n)) /
INR (fact (2 * S (N + n))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Rdiv; rewrite Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
Binomial.C (2 * S (N + n)) (S (N + n)) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Binomial.C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
(INR (fact (2 * S (N + n))) /
 (INR (fact (S (N + n))) *
  INR (fact (2 * S (N + n) - S (N + n))))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Rdiv; repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
INR (fact (2 * S (N + n))) *
/
(INR (fact (S (N + n))) *
 INR (fact (2 * S (N + n) - S (N + n)))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 *
/
(INR (fact (S (N + n))) *
 INR (fact (2 * S (N + n) - S (N + n)))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (S (N + n))) *
 INR (fact (2 * S (N + n) - S (N + n)))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ (INR (fact (S (N + n))) * INR (fact (S (N + n)))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + n) = (2 * S (N + n) - S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (S (N + n))) * / INR (fact (S (N + n))) =
(/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + n) = (2 * S (N + n) - S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Rsqr; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + n) = (2 * S (N + n) - S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + n) = (2 * S (N + n) - S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (N + n) = (2 * S (N + n) - S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) /
INR (fact (2 * S (N + n))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Rdiv; rewrite Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Binomial.C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
(INR (fact (2 * S (N + n))) /
 (INR (fact (2 * S (n0 + n))) *
  INR (fact (2 * S (N + n) - 2 * S (n0 + n))))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  unfold Rdiv; repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (N + n))) *
INR (fact (2 * S (N + n))) *
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * S (N + n) - 2 * S (n0 + n)))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 *
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * S (N + n) - 2 * S (n0 + n)))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * S (N + n) - 2 * S (n0 + n)))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * (N - n0)))) =
/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0))%nat =
(2 * S (N + n) - 2 * S (n0 + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * (N - n0)))) =
/
(INR (fact (2 * S (n0 + n))) *
 INR (fact (2 * (N - n0))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0))%nat =
(2 * S (N + n) - 2 * S (n0 + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0))%nat =
(2 * S (N + n) - 2 * S (n0 + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y (Init.Nat.pred N)
  apply Rle_trans with
    (sum_f_R0 (fun k:nat => INR N / INR (fact (S N)) * C ^ (4 * N)) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * N) * (/ INR (fact (S (N + k))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun _ : nat =>
   C ^ (4 * N) * (/ INR (fact (S (N + n))))²)
  (Init.Nat.pred (N - n)) <=
INR N / INR (fact (S N)) * C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <-
    (scal_sum (fun _:nat => C ^ (4 * N)) (pred (N - n))
      (Rsqr (/ INR (fact (S (N + n)))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
sum_f_R0 (fun _ : nat => C ^ (4 * N))
  (Init.Nat.pred (N - n)) <=
INR N / INR (fact (S N)) * C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite sum_cte.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
(C ^ (4 * N) * INR (S (Init.Nat.pred (N - n)))) <=
INR N / INR (fact (S N)) * C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² * C ^ (4 * N) *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S N)) * C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  do 2 rewrite <- (Rmult_comm (C ^ (4 * N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
C ^ (4 * N) * (/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
C ^ (4 * N) * (INR N / INR (fact (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
C ^ (4 * N) *
((/ INR (fact (S (N + n))))² *
 INR (S (Init.Nat.pred (N - n)))) <=
C ^ (4 * N) * (INR N / INR (fact (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  left; apply Rlt_le_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² *
INR (S (Init.Nat.pred (N - n))) <=
(/ INR (fact (S (N + n))))² * INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² * INR N <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= (/ INR (fact (S (N + n))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (S (Init.Nat.pred (N - n))) <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² * INR N <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_0_sqr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (S (Init.Nat.pred (N - n))) <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² * INR N <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(S (Init.Nat.pred (N - n)) <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² * INR N <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² * INR N <=
INR N / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply pos_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rle_trans with (/ INR (fact (S (N + n)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² <=
/ INR (fact (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  pattern (/ INR (fact (S (N + n)))) at 2; rewrite <- Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (N + n))))² <=
/ INR (fact (S (N + n))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  unfold Rsqr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) * / INR (fact (S (N + n))) <=
/ INR (fact (S (N + n))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= / INR (fact (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < INR (fact (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) * / INR (fact (S (N + n))) <=
INR (fact (S (N + n))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) * / INR (fact (S (N + n))) <=
INR (fact (S (N + n))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= INR (fact (S (N + n))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= INR (fact (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply (le_INR 1).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(1 <= fact (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply lt_le_S.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(0 < fact (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_lt; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (N + n))) <= / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_reg_l with (INR (fact (S (N + n)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < INR (fact (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) * / INR (fact (S (N + n))) <=
INR (fact (S (N + n))) * / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) * / INR (fact (S (N + n))) <=
INR (fact (S (N + n))) * / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= INR (fact (S (N + n))) * / INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply Rmult_le_reg_l with (INR (fact (S N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) * 1 <=
INR (fact (S N)) *
(INR (fact (S (N + n))) * / INR (fact (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) * 1 <=
INR (fact (S N)) *
(INR (fact (S (N + n))) * / INR (fact (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <=
INR (fact (S N)) *
(INR (fact (S (N + n))) * / INR (fact (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite (Rmult_comm (INR (fact (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <=
INR (fact (S (N + n))) * / INR (fact (S N)) *
INR (fact (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <=
INR (fact (S (N + n))) *
(/ INR (fact (S N)) * INR (fact (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <= INR (fact (S (N + n))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <= INR (fact (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(fact (S N) <= fact (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply fact_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(S N <= S (N + n))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_n_S.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(N <= N + n)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply le_plus_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (N + n))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S N)) * C ^ (4 * N))
  (Init.Nat.pred N) <= Majxy x y (Init.Nat.pred N)
  rewrite sum_cte.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N / INR (fact (S N)) * C ^ (4 * N) *
INR (S (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N / INR (fact (S N)) * C ^ (4 * N) *
INR (S (Init.Nat.pred N)) <=
C ^ (4 * N) / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite <- (Rmult_comm (C ^ (4 * N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) * (INR N / INR (fact (S N))) *
INR (S (Init.Nat.pred N)) <=
C ^ (4 * N) / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= C ^ (4 * N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S N)) * INR (S (Init.Nat.pred N)) <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S N)) * INR (S (Init.Nat.pred N)) <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  left; apply Rlt_le_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 < 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S N)) * INR (S (Init.Nat.pred N)) <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S N)) * INR (S (Init.Nat.pred N)) <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S N)) * INR (S (Init.Nat.pred N)) <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  cut (S (pred N) = N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N ->
INR N * / INR (fact (S N)) * INR (S (Init.Nat.pred N)) <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  intro; rewrite H0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / INR (fact (S N)) * INR N <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  pattern N at 2; rewrite <- H0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / INR (fact (S (S (Init.Nat.pred N)))) * INR N <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  do 2 rewrite fact_simpl.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
/
INR
  (S (S (Init.Nat.pred N)) *
   (S (Init.Nat.pred N) * fact (Init.Nat.pred N))) *
INR N <= / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite H0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / INR (S N * (N * fact (Init.Nat.pred N))) *
INR N <= / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  repeat rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
/ (INR (S N) * (INR N * INR (fact (Init.Nat.pred N)))) *
INR N <= / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  repeat rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
(/ INR (S N) *
 (/ INR N * / INR (fact (Init.Nat.pred N)))) * INR N <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite (Rmult_comm (/ INR (S N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
(/ INR N * / INR (fact (Init.Nat.pred N)) *
 / INR (S N)) * INR N <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / INR N * / INR (fact (Init.Nat.pred N)) *
/ INR (S N) * INR N <= / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
1 * / INR (fact (Init.Nat.pred N)) * / INR (S N) *
INR N <= / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (fact (Init.Nat.pred N)) * / INR (S N) * INR N <=
/ INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  pattern (/ INR (fact (pred N))) at 2; rewrite <- Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (fact (Init.Nat.pred N)) * / INR (S N) * INR N <=
/ INR (fact (Init.Nat.pred N)) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (fact (Init.Nat.pred N)) * (/ INR (S N) * INR N) <=
/ INR (fact (Init.Nat.pred N)) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * INR N <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * INR N <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply Rmult_le_reg_l with (INR (S N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 < INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * (/ INR (S N) * INR N) <= INR (S N) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply lt_INR_0; apply lt_O_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * (/ INR (S N) * INR N) <= INR (S N) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
1 * INR N <= INR (S N) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  rewrite Rmult_1_r; rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply le_INR; apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply not_O_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
N <> 0%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply not_O_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
N <> 0%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply prod_neq_R0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply not_O_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
N <> 0%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  red; intro; rewrite H1 in H; elim (lt_irrefl _ H).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact (Init.Nat.pred N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  symmetry ; apply S_pred with 0%nat; assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) <=
Majxy x y (Init.Nat.pred N)
  right.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) =
Majxy x y (Init.Nat.pred N)
  unfold Majxy.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * N) / INR (fact (Init.Nat.pred N)) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Init.Nat.pred N)) /
INR (fact (Init.Nat.pred N))
  unfold C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * N) /
INR (fact (Init.Nat.pred N)) =
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Init.Nat.pred N)) /
INR (fact (Init.Nat.pred N))
  replace (S (pred N)) with N.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * N) /
INR (fact (Init.Nat.pred N)) =
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * N) /
INR (fact (Init.Nat.pred N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
N = S (Init.Nat.pred N)
  apply S_pred with 0%nat; assumption.
Qed.

Lemma reste2_maj :
  forall (x y:R) (N:nat), (0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N.
forall (x y : R) (N : nat),
(0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N
Proof.
forall (x y : R) (N : nat),
(0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N
  intros.
x, y:R
N:nat
H:(0 < N)%nat
Rabs (Reste2 x y N) <= Majxy x y N
  set (C := Rmax 1 (Rmax (Rabs x) (Rabs y))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs (Reste2 x y N) <= Majxy x y N
  unfold Reste2.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs
  (sum_f_R0
     (fun k : nat =>
      sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1))
        (Init.Nat.pred (N - k))) (Init.Nat.pred N)) <=
Majxy x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	Rabs
        (sum_f_R0
          (fun l:nat =>
            (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
            x ^ (2 * S (l + k) + 1) *
            ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
            y ^ (2 * (N - l) + 1)) (pred (N - k)))) (
	      pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rabs
  (sum_f_R0
     (fun k : nat =>
      sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1))
        (Init.Nat.pred (N - k))) (Init.Nat.pred N)) <=
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N) <=
Majxy x y N
  apply
    (Rsum_abs
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
          x ^ (2 * S (l + k) + 1) *
          ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
          y ^ (2 * (N - l) + 1)) (pred (N - k))) (
	    pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          Rabs
          ((-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) *
            x ^ (2 * S (l + k) + 1) *
            ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
            y ^ (2 * (N - l) + 1))) (pred (N - k))) (
	      pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   Rabs
     (sum_f_R0
        (fun l : nat =>
         (-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1))
        (Init.Nat.pred (N - k)))) (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1)))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1)))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply sum_Rle.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
forall n : nat,
(n <= Init.Nat.pred N)%nat ->
Rabs
  (sum_f_R0
     (fun l : nat =>
      (-1) ^ S (l + n) /
      INR (fact (2 * S (l + n) + 1)) *
      x ^ (2 * S (l + n) + 1) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
      y ^ (2 * (N - l) + 1)) (Init.Nat.pred (N - n))) <=
sum_f_R0
  (fun l : nat =>
   Rabs
     ((-1) ^ S (l + n) /
      INR (fact (2 * S (l + n) + 1)) *
      x ^ (2 * S (l + n) + 1) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
      y ^ (2 * (N - l) + 1))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1)))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
Rabs
  (sum_f_R0
     (fun l : nat =>
      (-1) ^ S (l + n) /
      INR (fact (2 * S (l + n) + 1)) *
      x ^ (2 * S (l + n) + 1) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
      y ^ (2 * (N - l) + 1)) (Init.Nat.pred (N - n))) <=
sum_f_R0
  (fun l : nat =>
   Rabs
     ((-1) ^ S (l + n) /
      INR (fact (2 * S (l + n) + 1)) *
      x ^ (2 * S (l + n) + 1) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
      y ^ (2 * (N - l) + 1))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1)))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply
    (Rsum_abs
      (fun l:nat =>
	(-1) ^ S (l + n) / INR (fact (2 * S (l + n) + 1)) *
	x ^ (2 * S (l + n) + 1) *
	((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
	y ^ (2 * (N - l) + 1)) (pred (N - n))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1)))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          / INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) *
          C ^ (2 * S (S (N + k)))) (pred (N - k))) (
	    pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      Rabs
        ((-1) ^ S (l + k) /
         INR (fact (2 * S (l + k) + 1)) *
         x ^ (2 * S (l + k) + 1) *
         ((-1) ^ (N - l) /
          INR (fact (2 * (N - l) + 1))) *
         y ^ (2 * (N - l) + 1)))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   Rabs
     ((-1) ^ S (l + n) /
      INR (fact (2 * S (l + n) + 1)) *
      x ^ (2 * S (l + n) + 1) *
      ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) *
      y ^ (2 * (N - l) + 1))) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun l : nat =>
   /
   INR
     (fact (2 * S (l + n) + 1) *
      fact (2 * (N - l) + 1)) *
   C ^ (2 * S (S (N + n)))) (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs
  ((-1) ^ S (n0 + n) / INR (fact (2 * S (n0 + n) + 1)) *
   x ^ (2 * S (n0 + n) + 1) *
   ((-1) ^ (N - n0) / INR (fact (2 * (N - n0) + 1))) *
   y ^ (2 * (N - n0) + 1)) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  unfold Rdiv; repeat rewrite Rabs_mult.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs ((-1) ^ S (n0 + n)) *
Rabs (/ INR (fact (2 * S (n0 + n) + 1))) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
(Rabs ((-1) ^ (N - n0)) *
 Rabs (/ INR (fact (2 * (N - n0) + 1)))) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  do 2 rewrite pow_1_abs.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 * Rabs (/ INR (fact (2 * S (n0 + n) + 1))) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
(1 * Rabs (/ INR (fact (2 * (N - n0) + 1)))) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  do 2 rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (/ INR (fact (2 * S (n0 + n) + 1))) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
Rabs (/ INR (fact (2 * (N - n0) + 1))) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n) + 1)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
Rabs (/ INR (fact (2 * (N - n0) + 1))) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite (Rabs_right (/ INR (fact (2 * (N - n0) + 1)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
/ INR (fact (2 * (N - n0) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
/ INR (fact (2 * (N - n0) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * (N - n0) + 1))) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
/ INR (fact (2 * (N - n0) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/ INR (fact (2 * S (n0 + n) + 1)) *
/ INR (fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) *
(Rabs (x ^ (2 * S (n0 + n) + 1)) *
 (/ INR (fact (2 * (N - n0) + 1)) *
  Rabs (y ^ (2 * (N - n0) + 1)))) <=
/ INR (fact (2 * S (n0 + n) + 1)) *
(/ INR (fact (2 * (N - n0) + 1)) *
 C ^ (2 * S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= / INR (fact (2 * S (n0 + n) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n) + 1)) *
(/ INR (fact (2 * (N - n0) + 1)) *
 Rabs (y ^ (2 * (N - n0) + 1))) <=
/ INR (fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n) + 1)) *
(/ INR (fact (2 * (N - n0) + 1)) *
 Rabs (y ^ (2 * (N - n0) + 1))) <=
/ INR (fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n) + 1)) *
/ INR (fact (2 * (N - n0) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/ INR (fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0) + 1)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) *
Rabs (x ^ (2 * S (n0 + n) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
/ INR (fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) *
(Rabs (x ^ (2 * S (n0 + n) + 1)) *
 Rabs (y ^ (2 * (N - n0) + 1))) <=
/ INR (fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= / INR (fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs (x ^ (2 * S (n0 + n) + 1)) *
Rabs (y ^ (2 * (N - n0) + 1)) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  do 2 rewrite <- RPow_abs.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) *
Rabs y ^ (2 * (N - n0) + 1) <= C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with (Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) *
Rabs y ^ (2 * (N - n0) + 1) <=
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs x ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y ^ (2 * (N - n0) + 1) <= C ^ (2 * (N - n0) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply pow_le; apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y ^ (2 * (N - n0) + 1) <= C ^ (2 * (N - n0) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply pow_incr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs y <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  split.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs y
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  unfold C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs y <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with (C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0) + 1))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * (N - n0) + 1) * Rabs x ^ (2 * S (n0 + n) + 1) <=
C ^ (2 * (N - n0) + 1) * C ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C ^ (2 * (N - n0) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) <=
C ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) <=
C ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) <=
C ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  left; apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) <=
C ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x ^ (2 * S (n0 + n) + 1) <=
C ^ (2 * S (n0 + n) + 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply pow_incr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs x <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  split.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= Rabs x
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rabs_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  unfold C; apply Rle_trans with (Rmax (Rabs x) (Rabs y)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rabs x <= Rmax (Rabs x) (Rabs y)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rmax (Rabs x) (Rabs y) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Rmax (Rabs x) (Rabs y) <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply RmaxLess2.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) <=
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  right.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) =
C ^ (2 * S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  replace (2 * S (S (N + n)))%nat with
    (2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1) =
C ^ (2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat =
(2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  repeat rewrite pow_add.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (n0 + n)) * C ^ 1 *
(C ^ (2 * (N - n0)) * C ^ 1) =
C ^ (2 * (N - n0)) * C ^ 1 *
(C ^ (2 * S (n0 + n)) * C ^ 1)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat =
(2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  ring.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat =
(2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (n0 + n) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * (N - n0) + 1)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * (N - n0) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_ge; left; apply Rinv_0_lt_compat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 < INR (fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (n0 + n) + 1)) >= 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_ge; left; apply Rinv_0_lt_compat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 < INR (fact (2 * S (n0 + n) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat =>
          / INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) *
          C ^ (4 * S N)) (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) *
      C ^ (2 * S (S (N + k)))) (Init.Nat.pred (N - k)))
  (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   /
   INR
     (fact (2 * S (l + n) + 1) *
      fact (2 * (N - l) + 1)) *
   C ^ (2 * S (S (N + n)))) (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun l : nat =>
   /
   INR
     (fact (2 * S (l + n) + 1) *
      fact (2 * (N - l) + 1)) * C ^ (4 * S N))
  (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (2 * S (S (N + n))) <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <=
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (S (N + n))) <= C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  left; apply Rinv_0_lt_compat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (S (N + n))) <= C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (2 * S (S (N + n))) <= C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rle_pow.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (N + n)) <= 4 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (N + n)) <= 4 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  replace (4 * S N)%nat with (2 * (2 * S N))%nat; [ idtac | ring ].
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (N + n)) <= 2 * (2 * S N))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply (fun m n p:nat => mult_le_compat_l p n m).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(S (S (N + n)) <= 2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  replace (2 * S N)%nat with (S (S (N + N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(S (S (N + n)) <= S (S (N + N)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + N)) = (2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  repeat apply le_n_S.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(N + n <= N + N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + N)) = (2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply plus_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + N)) = (2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply le_trans with (pred N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(n <= Init.Nat.pred N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + N)) = (2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(Init.Nat.pred N <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + N)) = (2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply le_pred_n.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + N)) = (2 * S N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  ring.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rle_trans with
    (sum_f_R0
      (fun k:nat =>
	sum_f_R0
        (fun l:nat => C ^ (4 * S N) * Rsqr (/ INR (fact (S (S (N + k))))))
        (pred (N - k))) (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun l : nat =>
      /
      INR
        (fact (2 * S (l + k) + 1) *
         fact (2 * (N - l) + 1)) * C ^ (4 * S N))
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun l : nat =>
   /
   INR
     (fact (2 * S (l + n) + 1) *
      fact (2 * (N - l) + 1)) * C ^ (4 * S N))
  (Init.Nat.pred (N - n)) <=
sum_f_R0
  (fun _ : nat =>
   C ^ (4 * S N) * (/ INR (fact (S (S (N + n)))))²)
  (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) *
C ^ (4 * S N) <=
C ^ (4 * S N) * (/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite <- (Rmult_comm (C ^ (4 * S N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
C ^ (4 * S N) *
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) <=
C ^ (4 * S N) * (/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  left; apply Rlt_le_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 < 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1)) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  replace (/ INR (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))) with
    (Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
      INR (fact (2 * S (S (N + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rle_trans with
    (Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
      INR (fact (2 * S (S (N + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) <=
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Rdiv;
    do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (S (N + n)))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) <=
/ INR (fact (2 * S (S (N + n)))) *
Binomial.C (2 * S (S (N + n))) (S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
0 <= / INR (fact (2 * S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) <=
Binomial.C (2 * S (S (N + n))) (S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) <=
Binomial.C (2 * S (S (N + n))) (S (S (N + n)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply C_maj.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (n0 + n) + 1 <= 2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply le_trans with (2 * S (S (n0 + n)))%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (n0 + n) + 1 <= 2 * S (S (n0 + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (n0 + n)) <= 2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (n0 + n) + 1 <= S (2 * S (n0 + n) + 1))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (2 * S (n0 + n) + 1) = (2 * S (S (n0 + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (n0 + n)) <= 2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (2 * S (n0 + n) + 1) = (2 * S (S (n0 + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (n0 + n)) <= 2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  ring.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * S (S (n0 + n)) <= 2 * S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) <=
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  right.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) /
INR (fact (2 * S (S (N + n)))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Rdiv; rewrite Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
Binomial.C (2 * S (S (N + n))) (S (S (N + n))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Binomial.C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
(INR (fact (2 * S (S (N + n)))) /
 (INR (fact (S (S (N + n)))) *
  INR (fact (2 * S (S (N + n)) - S (S (N + n)))))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Rdiv; repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
INR (fact (2 * S (S (N + n)))) *
/
(INR (fact (S (S (N + n)))) *
 INR (fact (2 * S (S (N + n)) - S (S (N + n))))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 *
/
(INR (fact (S (S (N + n)))) *
 INR (fact (2 * S (S (N + n)) - S (S (N + n))))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (S (S (N + n)))) *
 INR (fact (2 * S (S (N + n)) - S (S (N + n))))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  replace (2 * S (S (N + n)) - S (S (N + n)))%nat with (S (S (N + n))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (S (S (N + n)))) *
 INR (fact (S (S (N + n))))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + n)) =
(2 * S (S (N + n)) - S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (S (S (N + n)))) *
/ INR (fact (S (S (N + n)))) =
(/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + n)) =
(2 * S (S (N + n)) - S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Rsqr; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + n)) =
(2 * S (S (N + n)) - S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + n)) =
(2 * S (S (N + n)) - S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
S (S (N + n)) =
(2 * S (S (N + n)) - S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) /
INR (fact (2 * S (S (N + n)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Rdiv; rewrite Rmult_comm.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Binomial.C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
(INR (fact (2 * S (S (N + n)))) /
 (INR (fact (2 * S (n0 + n) + 1)) *
  INR
    (fact (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  unfold Rdiv; repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/ INR (fact (2 * S (S (N + n)))) *
INR (fact (2 * S (S (N + n)))) *
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * S (S (N + n)) - (2 * S (n0 + n) + 1)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
1 *
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * S (S (N + n)) - (2 * S (n0 + n) + 1)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * S (S (N + n)) - (2 * S (n0 + n) + 1)))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  replace (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat with
    (2 * (N - n0) + 1)%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * (N - n0) + 1))) =
/
INR
  (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 1)%nat =
(2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * (N - n0) + 1))) =
/
(INR (fact (2 * S (n0 + n) + 1)) *
 INR (fact (2 * (N - n0) + 1)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 1)%nat =
(2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
(2 * (N - n0) + 1)%nat =
(2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
n0:nat
H1:(n0 <= Init.Nat.pred (N - n))%nat
INR (fact (2 * S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
Majxy x y N
  apply Rle_trans with
    (sum_f_R0 (fun k:nat => INR N / INR (fact (S (S N))) * C ^ (4 * S N))
      (pred N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun k : nat =>
   sum_f_R0
     (fun _ : nat =>
      C ^ (4 * S N) * (/ INR (fact (S (S (N + k)))))²)
     (Init.Nat.pred (N - k))) (Init.Nat.pred N) <=
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply sum_Rle; intros.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
sum_f_R0
  (fun _ : nat =>
   C ^ (4 * S N) * (/ INR (fact (S (S (N + n)))))²)
  (Init.Nat.pred (N - n)) <=
INR N / INR (fact (S (S N))) * C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <-
    (scal_sum (fun _:nat => C ^ (4 * S N)) (pred (N - n))
      (Rsqr (/ INR (fact (S (S (N + n))))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
sum_f_R0 (fun _ : nat => C ^ (4 * S N))
  (Init.Nat.pred (N - n)) <=
INR N / INR (fact (S (S N))) * C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite sum_cte.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
(C ^ (4 * S N) * INR (S (Init.Nat.pred (N - n)))) <=
INR N / INR (fact (S (S N))) * C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * C ^ (4 * S N) *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S (S N))) * C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  do 2 rewrite <- (Rmult_comm (C ^ (4 * S N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
C ^ (4 * S N) * (/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
C ^ (4 * S N) * (INR N / INR (fact (S (S N))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
C ^ (4 * S N) *
((/ INR (fact (S (S (N + n)))))² *
 INR (S (Init.Nat.pred (N - n)))) <=
C ^ (4 * S N) * (INR N / INR (fact (S (S N))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  left; apply Rlt_le_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with (Rsqr (/ INR (fact (S (S (N + n))))) * INR N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² *
INR (S (Init.Nat.pred (N - n))) <=
(/ INR (fact (S (S (N + n)))))² * INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= (/ INR (fact (S (S (N + n)))))²
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (S (Init.Nat.pred (N - n))) <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_0_sqr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (S (Init.Nat.pred (N - n))) <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  replace (S (pred (N - n))) with (N - n)%nat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (N - n) <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(N - n <= N)%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(N - n)%nat = S (Init.Nat.pred (N - n))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² * INR N <=
INR N / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_comm; unfold Rdiv; apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² <=
/ INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply pos_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² <=
/ INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rle_trans with (/ INR (fact (S (S (N + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² <=
/ INR (fact (S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  pattern (/ INR (fact (S (S (N + n))))) at 2; rewrite <- Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(/ INR (fact (S (S (N + n)))))² <=
/ INR (fact (S (S (N + n)))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  unfold Rsqr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) *
/ INR (fact (S (S (N + n)))) <=
/ INR (fact (S (S (N + n)))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 <= / INR (fact (S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < INR (fact (S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) *
/ INR (fact (S (S (N + n)))) <=
INR (fact (S (S (N + n)))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) *
/ INR (fact (S (S (N + n)))) <=
INR (fact (S (S (N + n)))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= INR (fact (S (S (N + n)))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <= INR (fact (S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply (le_INR 1).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(1 <= fact (S (S (N + n))))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply lt_le_S.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(0 < fact (S (S (N + n))))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_lt; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
/ INR (fact (S (S (N + n)))) <= / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < INR (fact (S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) *
/ INR (fact (S (S (N + n)))) <=
INR (fact (S (S (N + n)))) * / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) *
/ INR (fact (S (S (N + n)))) <=
INR (fact (S (S (N + n)))) * / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
1 <=
INR (fact (S (S (N + n)))) * / INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply Rmult_le_reg_l with (INR (fact (S (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
0 < INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) * 1 <=
INR (fact (S (S N))) *
(INR (fact (S (S (N + n)))) * / INR (fact (S (S N))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) * 1 <=
INR (fact (S (S N))) *
(INR (fact (S (S (N + n)))) * / INR (fact (S (S N))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <=
INR (fact (S (S N))) *
(INR (fact (S (S (N + n)))) * / INR (fact (S (S N))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite (Rmult_comm (INR (fact (S (S N))))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <=
INR (fact (S (S (N + n)))) * / INR (fact (S (S N))) *
INR (fact (S (S N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <=
INR (fact (S (S (N + n)))) *
(/ INR (fact (S (S N))) * INR (fact (S (S N))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <= INR (fact (S (S (N + n)))) * 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite Rmult_1_r.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <= INR (fact (S (S (N + n))))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(fact (S (S N)) <= fact (S (S (N + n))))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply fact_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
(S (S N) <= S (S (N + n)))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  omega.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S N))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
n:nat
H0:(n <= Init.Nat.pred N)%nat
INR (fact (S (S (N + n)))) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
sum_f_R0
  (fun _ : nat =>
   INR N / INR (fact (S (S N))) * C ^ (4 * S N))
  (Init.Nat.pred N) <= Majxy x y N
  rewrite sum_cte.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N / INR (fact (S (S N))) * C ^ (4 * S N) *
INR (S (Init.Nat.pred N)) <= Majxy x y N
  apply Rle_trans with (C ^ (4 * S N) / INR (fact N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N / INR (fact (S (S N))) * C ^ (4 * S N) *
INR (S (Init.Nat.pred N)) <=
C ^ (4 * S N) / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite <- (Rmult_comm (C ^ (4 * S N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) * (INR N / INR (fact (S (S N)))) *
INR (S (Init.Nat.pred N)) <=
C ^ (4 * S N) / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  unfold Rdiv; rewrite Rmult_assoc; apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= C ^ (4 * S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S (S N))) *
INR (S (Init.Nat.pred N)) <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply pow_le.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S (S N))) *
INR (S (Init.Nat.pred N)) <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  left; apply Rlt_le_trans with 1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
0 < 1
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S (S N))) *
INR (S (Init.Nat.pred N)) <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply Rlt_0_1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
1 <= C
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S (S N))) *
INR (S (Init.Nat.pred N)) <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  unfold C; apply RmaxLess1.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
INR N * / INR (fact (S (S N))) *
INR (S (Init.Nat.pred N)) <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  cut (S (pred N) = N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N ->
INR N * / INR (fact (S (S N))) *
INR (S (Init.Nat.pred N)) <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  intro; rewrite H0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / INR (fact (S (S N))) * INR N <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  do 2 rewrite fact_simpl.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / INR (S (S N) * (S N * fact N)) * INR N <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat rewrite mult_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N * / (INR (S (S N)) * (INR (S N) * INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat rewrite Rinv_mult_distr.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply Rle_trans with
    (INR (S (S N)) * (/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) * INR N).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <=
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N *
(/ INR (S (S N)) *
 (/ INR (S N) * (/ INR (fact N) * INR N))) <=
INR (S (S N)) *
(/ INR (S (S N)) *
 (/ INR (S N) * (/ INR (fact N) * INR N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite (Rmult_comm (INR N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S (S N)) *
(/ INR (S N) * (/ INR (fact N) * INR N)) * INR N <=
INR (S (S N)) *
(/ INR (S (S N)) *
 (/ INR (S N) * (/ INR (fact N) * INR N)))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite (Rmult_comm (INR (S (S N)))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S (S N)) *
(/ INR (S N) * (/ INR (fact N) * INR N)) * INR N <=
/ INR (S (S N)) *
(/ INR (S N) * (/ INR (fact N) * INR N)) *
INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <=
/ INR (S (S N)) *
(/ INR (S N) * (/ INR (fact N) * INR N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat apply Rmult_le_pos.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  left; apply Rinv_0_lt_compat.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 < INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= INR N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply pos_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply le_INR.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
(N <= S (S N))%nat
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply le_trans with (S N); apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) *
(/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) *
INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat rewrite <- Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) * / INR (S (S N)) * / INR (S N) *
/ INR (fact N) * INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite <- Rinv_r_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
1 * / INR (S N) * / INR (fact N) * INR N <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite Rmult_1_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR N <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply Rle_trans with (/ INR (S N) * / INR (fact N) * INR (S N)).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR N <=
/ INR (S N) * / INR (fact N) * INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * (/ INR (fact N) * INR N) <=
/ INR (S N) * (/ INR (fact N) * INR (S N))
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  repeat apply Rmult_le_compat_l.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
0 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR N <= INR (S N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply le_INR; apply le_n_Sn.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (S N) * / INR (fact N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite (Rmult_comm (/ INR (S N))).
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (fact N) * / INR (S N) * INR (S N) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite Rmult_assoc.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (fact N) * (/ INR (S N) * INR (S N)) <=
/ INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite <- Rinv_l_sym.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
/ INR (fact N) * 1 <= / INR (fact N)
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  rewrite Rmult_1_r; right; reflexivity.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply INR_fact_neq_0.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S (S N)) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply not_O_INR; discriminate.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
H0:S (Init.Nat.pred N) = N
INR (S N) * INR (fact N) <> 0
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ].
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
S (Init.Nat.pred N) = N
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  symmetry ; apply S_pred with 0%nat; assumption.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) <= Majxy x y N
  right.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) = Majxy x y N
  unfold Majxy.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
C ^ (4 * S N) / INR (fact N) =
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S N) /
INR (fact N)
  unfold C.
x, y:R
N:nat
H:(0 < N)%nat
C:=Rmax 1 (Rmax (Rabs x) (Rabs y)):R
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S N) /
INR (fact N) =
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S N) /
INR (fact N)
  reflexivity.
Qed.

Lemma reste1_cv_R0 : forall x y:R, Un_cv (Reste1 x y) 0.
forall x y : R, Un_cv (Reste1 x y) 0
Proof.
forall x y : R, Un_cv (Reste1 x y) 0
  intros.
x, y:R
Un_cv (Reste1 x y) 0
  assert (H := Majxy_cv_R0 x y).
x, y:R
H:Un_cv (Majxy x y) 0
Un_cv (Reste1 x y) 0
  unfold Un_cv in H; unfold R_dist in H.
x, y:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps
Un_cv (Reste1 x y) 0
  unfold Un_cv; unfold R_dist; intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps0
eps:R
H0:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y n - 0) < eps
  elim (H eps H0); intros N0 H1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n : nat,
(n >= N0)%nat -> Rabs (Majxy x y n - 0) < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y n - 0) < eps
  exists (S N0); intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste1 x y n - 0) < eps
  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste1 x y n) < eps
  apply Rle_lt_trans with (Rabs (Majxy x y (pred n))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste1 x y n) <=
Rabs (Majxy x y (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  rewrite (Rabs_right (Majxy x y (pred n))).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste1 x y n) <= Majxy x y (Init.Nat.pred n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y (Init.Nat.pred n) >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply reste1_maj.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(0 < n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y (Init.Nat.pred n) >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply lt_le_trans with (S N0).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(0 < S N0)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y (Init.Nat.pred n) >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply lt_O_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y (Init.Nat.pred n) >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  assumption.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y (Init.Nat.pred n) >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply Rle_ge.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= Majxy x y (Init.Nat.pred n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  unfold Majxy.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Init.Nat.pred n)) /
INR (fact (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  unfold Rdiv; apply Rmult_le_pos.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y))
^ (4 * S (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply pow_le.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply Rle_trans with 1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  left; apply Rlt_0_1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  apply RmaxLess1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact (Init.Nat.pred n))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n)) < eps
  replace (Majxy x y (pred n)) with (Majxy x y (pred n) - 0); [ idtac | ring ].
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y (Init.Nat.pred n) - 0) < eps
  apply H1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(Init.Nat.pred n >= N0)%nat
  unfold ge; apply le_S_n.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= S (Init.Nat.pred n))%nat
  replace (S (pred n)) with n.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
n = S (Init.Nat.pred n)
  assumption.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
n = S (Init.Nat.pred n)
  apply S_pred with 0%nat.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(0 < n)%nat
  apply lt_le_trans with (S N0); [ apply lt_O_Sn | assumption ].
Qed.

Lemma reste2_cv_R0 : forall x y:R, Un_cv (Reste2 x y) 0.
forall x y : R, Un_cv (Reste2 x y) 0
Proof.
forall x y : R, Un_cv (Reste2 x y) 0
  intros.
x, y:R
Un_cv (Reste2 x y) 0
  assert (H := Majxy_cv_R0 x y).
x, y:R
H:Un_cv (Majxy x y) 0
Un_cv (Reste2 x y) 0
  unfold Un_cv in H; unfold R_dist in H.
x, y:R
H:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps
Un_cv (Reste2 x y) 0
  unfold Un_cv; unfold R_dist; intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps0
eps:R
H0:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste2 x y n - 0) < eps
  elim (H eps H0); intros N0 H1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Majxy x y n - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n : nat,
(n >= N0)%nat -> Rabs (Majxy x y n - 0) < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste2 x y n - 0) < eps
  exists (S N0); intros.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste2 x y n - 0) < eps
  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste2 x y n) < eps
  apply Rle_lt_trans with (Rabs (Majxy x y n)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste2 x y n) <= Rabs (Majxy x y n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  rewrite (Rabs_right (Majxy x y n)).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Reste2 x y n) <= Majxy x y n
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y n >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply reste2_maj.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(0 < n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y n >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply lt_le_trans with (S N0).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(0 < S N0)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y n >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply lt_O_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y n >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  assumption.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Majxy x y n >= 0
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply Rle_ge.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= Majxy x y n
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  unfold Majxy.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <=
Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) /
INR (fact n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  unfold Rdiv; apply Rmult_le_pos.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply pow_le.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply Rle_trans with 1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= 1
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  left; apply Rlt_0_1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
1 <= Rmax 1 (Rmax (Rabs x) (Rabs y))
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  apply RmaxLess1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
0 <= / INR (fact n)
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  left; apply Rinv_0_lt_compat; apply INR_fact_lt_0.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n) < eps
  replace (Majxy x y n) with (Majxy x y n - 0); [ idtac | ring ].
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
Rabs (Majxy x y n - 0) < eps
  apply H1.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(n >= N0)%nat
  unfold ge; apply le_trans with (S N0).
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(N0 <= S N0)%nat
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
  apply le_n_Sn.
x, y:R
H:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat -> Rabs (Majxy x y n0 - 0) < eps0
eps:R
H0:eps > 0
N0:nat
H1:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Majxy x y n0 - 0) < eps
n:nat
H2:(n >= S N0)%nat
(S N0 <= n)%nat
  exact H2.
Qed.

Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0.
forall x y : R, Un_cv (Reste x y) 0
Proof.
forall x y : R, Un_cv (Reste x y) 0
  intros.
x, y:R
Un_cv (Reste x y) 0
  unfold Reste.
x, y:R
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  set (An := fun n:nat => Reste2 x y n).
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Un_cv (fun N : nat => An N - Reste1 x y (S N)) 0
  set (Bn := fun n:nat => Reste1 x y (S n)).
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun N : nat => An N - Reste1 x y (S N)) 0
  cut
    (Un_cv (fun n:nat => An n - Bn n) (0 - 0) ->
      Un_cv (fun N:nat => Reste2 x y N - Reste1 x y (S N)) 0).
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
(Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
 Un_cv
   (fun N : nat => Reste2 x y N - Reste1 x y (S N)) 0) ->
Un_cv (fun N : nat => An N - Reste1 x y (S N)) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  intro.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv (fun N : nat => An N - Reste1 x y (S N)) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  apply H.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv (fun n : nat => An n - Bn n) (0 - 0)
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  apply CV_minus.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv An 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv Bn 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  unfold An.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv (fun n : nat => Reste2 x y n) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv Bn 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  replace (fun n:nat => Reste2 x y n) with (Reste2 x y).
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv (Reste2 x y) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Reste2 x y = (fun n : nat => Reste2 x y n)
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv Bn 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  apply reste2_cv_R0.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Reste2 x y = (fun n : nat => Reste2 x y n)
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv Bn 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  reflexivity.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv Bn 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  unfold Bn.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
Un_cv (fun n : nat => Reste1 x y (S n)) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  assert (H0 := reste1_cv_R0 x y).
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:Un_cv (Reste1 x y) 0
Un_cv (fun n : nat => Reste1 x y (S n)) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  unfold Un_cv in H0; unfold R_dist in H0.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y n - 0) < eps
Un_cv (fun n : nat => Reste1 x y (S n)) 0
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  unfold Un_cv; unfold R_dist; intros.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y n - 0) < eps0
eps:R
H1:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y (S n) - 0) < eps
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  elim (H0 eps H1); intros N0 H2.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y n - 0) < eps0
eps:R
H1:eps > 0
N0:nat
H2:forall n : nat,
(n >= N0)%nat -> Rabs (Reste1 x y n - 0) < eps
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste1 x y (S n) - 0) < eps
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  exists N0; intros.
x, y:R
An:=fun n0 : nat => Reste2 x y n0:nat -> R
Bn:=fun n0 : nat => Reste1 x y (S n0):nat -> R
H:Un_cv (fun n0 : nat => An n0 - Bn n0) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (Reste1 x y n0 - 0) < eps0
eps:R
H1:eps > 0
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Reste1 x y n0 - 0) < eps
n:nat
H3:(n >= N0)%nat
Rabs (Reste1 x y (S n) - 0) < eps
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  apply H2.
x, y:R
An:=fun n0 : nat => Reste2 x y n0:nat -> R
Bn:=fun n0 : nat => Reste1 x y (S n0):nat -> R
H:Un_cv (fun n0 : nat => An n0 - Bn n0) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (Reste1 x y n0 - 0) < eps0
eps:R
H1:eps > 0
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Reste1 x y n0 - 0) < eps
n:nat
H3:(n >= N0)%nat
(S n >= N0)%nat
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  unfold ge; apply le_trans with (S N0).
x, y:R
An:=fun n0 : nat => Reste2 x y n0:nat -> R
Bn:=fun n0 : nat => Reste1 x y (S n0):nat -> R
H:Un_cv (fun n0 : nat => An n0 - Bn n0) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (Reste1 x y n0 - 0) < eps0
eps:R
H1:eps > 0
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Reste1 x y n0 - 0) < eps
n:nat
H3:(n >= N0)%nat
(N0 <= S N0)%nat
x, y:R
An:=fun n0 : nat => Reste2 x y n0:nat -> R
Bn:=fun n0 : nat => Reste1 x y (S n0):nat -> R
H:Un_cv (fun n0 : nat => An n0 - Bn n0) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (Reste1 x y n0 - 0) < eps0
eps:R
H1:eps > 0
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Reste1 x y n0 - 0) < eps
n:nat
H3:(n >= N0)%nat
(S N0 <= S n)%nat
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  apply le_n_Sn.
x, y:R
An:=fun n0 : nat => Reste2 x y n0:nat -> R
Bn:=fun n0 : nat => Reste1 x y (S n0):nat -> R
H:Un_cv (fun n0 : nat => An n0 - Bn n0) (0 - 0) ->
Un_cv
  (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
H0:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n0 : nat,
  (n0 >= N)%nat ->
  Rabs (Reste1 x y n0 - 0) < eps0
eps:R
H1:eps > 0
N0:nat
H2:forall n0 : nat,
(n0 >= N0)%nat -> Rabs (Reste1 x y n0 - 0) < eps
n:nat
H3:(n >= N0)%nat
(S N0 <= S n)%nat
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  apply le_n_S; assumption.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => An n - Bn n) (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  unfold An, Bn.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
Un_cv (fun n : nat => Reste2 x y n - Reste1 x y (S n))
  (0 - 0) ->
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  intro.
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv
  (fun n : nat => Reste2 x y n - Reste1 x y (S n))
  (0 - 0)
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  0
  replace 0 with (0 - 0); [ idtac | ring ].
x, y:R
An:=fun n : nat => Reste2 x y n:nat -> R
Bn:=fun n : nat => Reste1 x y (S n):nat -> R
H:Un_cv
  (fun n : nat => Reste2 x y n - Reste1 x y (S n))
  (0 - 0)
Un_cv (fun N : nat => Reste2 x y N - Reste1 x y (S N))
  (0 - 0)
  exact H.
Qed.

Theorem cos_plus : forall x y:R, cos (x + y) = cos x * cos y - sin x * sin y.
forall x y : R,
cos (x + y) = cos x * cos y - sin x * sin y
Proof.
forall x y : R,
cos (x + y) = cos x * cos y - sin x * sin y
  intros.
x, y:R
cos (x + y) = cos x * cos y - sin x * sin y
  cut (Un_cv (C1 x y) (cos x * cos y - sin x * sin y)).
x, y:R
Un_cv (C1 x y) (cos x * cos y - sin x * sin y) ->
cos (x + y) = cos x * cos y - sin x * sin y
x, y:R
Un_cv (C1 x y) (cos x * cos y - sin x * sin y)
  cut (Un_cv (C1 x y) (cos (x + y))).
x, y:R
Un_cv (C1 x y) (cos (x + y)) ->
Un_cv (C1 x y) (cos x * cos y - sin x * sin y) ->
cos (x + y) = cos x * cos y - sin x * sin y
x, y:R
Un_cv (C1 x y) (cos (x + y))
x, y:R
Un_cv (C1 x y) (cos x * cos y - sin x * sin y)
  intros.
x, y:R
H:Un_cv (C1 x y) (cos (x + y))
H0:Un_cv (C1 x y) (cos x * cos y - sin x * sin y)
cos (x + y) = cos x * cos y - sin x * sin y
x, y:R
Un_cv (C1 x y) (cos (x + y))
x, y:R
Un_cv (C1 x y) (cos x * cos y - sin x * sin y)
  apply UL_sequence with (C1 x y); assumption.
x, y:R
Un_cv (C1 x y) (cos (x + y))
x, y:R
Un_cv (C1 x y) (cos x * cos y - sin x * sin y)
  apply C1_cvg.
x, y:R
Un_cv (C1 x y) (cos x * cos y - sin x * sin y)
  unfold Un_cv; unfold R_dist.
x, y:R
forall eps : R,
eps > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  intros.
x, y, eps:R
H:eps > 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H0 := A1_cvg x).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H1 := A1_cvg y).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H2 := B1_cvg x).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H3 := B1_cvg y).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H4 := CV_mult _ _ _ _ H0 H1).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:Un_cv (fun i : nat => A1 x i * A1 y i)
  (cos x * cos y)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H5 := CV_mult _ _ _ _ H2 H3).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:Un_cv (fun i : nat => A1 x i * A1 y i)
  (cos x * cos y)
H5:Un_cv (fun i : nat => B1 x i * B1 y i)
  (sin x * sin y)
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  assert (H6 := reste_cv_R0 x y).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:Un_cv (fun i : nat => A1 x i * A1 y i)
  (cos x * cos y)
H5:Un_cv (fun i : nat => B1 x i * B1 y i)
  (sin x * sin y)
H6:Un_cv (Reste x y) 0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  unfold Un_cv in H4; unfold Un_cv in H5; unfold Un_cv in H6.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (A1 x n * A1 y n) (cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  R_dist (B1 x n * B1 y n) (sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> R_dist (Reste x y n) 0 < eps0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  unfold R_dist in H4; unfold R_dist in H5; unfold R_dist in H6.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste x y n - 0) < eps0
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  cut (0 < eps / 3);
    [ intro
      | unfold Rdiv; apply Rmult_lt_0_compat;
	[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste x y n - 0) < eps0
H7:0 < eps / 3
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  elim (H4 (eps / 3) H7); intros N1 H8.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste x y n - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n : nat,
(n >= N1)%nat ->
Rabs (A1 x n * A1 y n - cos x * cos y) < eps / 3
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  elim (H5 (eps / 3) H7); intros N2 H9.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste x y n - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n : nat,
(n >= N1)%nat ->
Rabs (A1 x n * A1 y n - cos x * cos y) < eps / 3
N2:nat
H9:forall n : nat,
(n >= N2)%nat ->
Rabs (B1 x n * B1 y n - sin x * sin y) < eps / 3
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  elim (H6 (eps / 3) H7); intros N3 H10.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N : nat,
  forall n : nat,
  (n >= N)%nat -> Rabs (Reste x y n - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n : nat,
(n >= N1)%nat ->
Rabs (A1 x n * A1 y n - cos x * cos y) < eps / 3
N2:nat
H9:forall n : nat,
(n >= N2)%nat ->
Rabs (B1 x n * B1 y n - sin x * sin y) < eps / 3
N3:nat
H10:forall n : nat, (n >= N3)%nat -> Rabs (Reste x y n - 0) < eps / 3
exists N : nat,
  forall n : nat,
  (n >= N)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  set (N := S (S (max (max N1 N2) N3))).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (Reste x y n - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n : nat,
(n >= N1)%nat ->
Rabs (A1 x n * A1 y n - cos x * cos y) < eps / 3
N2:nat
H9:forall n : nat,
(n >= N2)%nat ->
Rabs (B1 x n * B1 y n - sin x * sin y) < eps / 3
N3:nat
H10:forall n : nat, (n >= N3)%nat -> Rabs (Reste x y n - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
  eps
  exists N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (A1 x n * A1 y n - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat ->
  Rabs (B1 x n * B1 y n - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n : nat,
  (n >= N0)%nat -> Rabs (Reste x y n - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n : nat,
(n >= N1)%nat ->
Rabs (A1 x n * A1 y n - cos x * cos y) < eps / 3
N2:nat
H9:forall n : nat,
(n >= N2)%nat ->
Rabs (B1 x n * B1 y n - sin x * sin y) < eps / 3
N3:nat
H10:forall n : nat, (n >= N3)%nat -> Rabs (Reste x y n - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
forall n : nat,
(n >= N)%nat ->
Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
eps
  intros.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
eps
  cut (n = S (pred n)).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n) ->
Rabs (C1 x y n - (cos x * cos y - sin x * sin y)) <
eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  intro; rewrite H12.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (C1 x y (S (Init.Nat.pred n)) -
   (cos x * cos y - sin x * sin y)) < eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- cos_plus_form.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (A1 x (S (Init.Nat.pred n)) *
   A1 y (S (Init.Nat.pred n)) -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n) -
   (cos x * cos y - sin x * sin y)) < eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- H12.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (A1 x n * A1 y n -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n) -
   (cos x * cos y - sin x * sin y)) < eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply Rle_lt_trans with
    (Rabs (A1 x n * A1 y n - cos x * cos y) +
      Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n))).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (A1 x n * A1 y n -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n) -
   (cos x * cos y - sin x * sin y)) <=
Rabs (A1 x n * A1 y n - cos x * cos y) +
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n))
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (A1 x n * A1 y n - cos x * cos y) +
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  replace
    (A1 x n * A1 y n - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n) -
      (cos x * cos y - sin x * sin y)) with
    (A1 x n * A1 y n - cos x * cos y +
      (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n)));
    [ apply Rabs_triang | ring ].
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (A1 x n * A1 y n - cos x * cos y) +
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  replace eps with (eps / 3 + (eps / 3 + eps / 3)).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (A1 x n * A1 y n - cos x * cos y) +
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) <
eps / 3 + (eps / 3 + eps / 3)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply Rplus_lt_compat.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (A1 x n * A1 y n - cos x * cos y) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply H8.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(n >= N1)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold ge; apply le_trans with N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N1 <= N)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N1 <= S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with (max N1 N2).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N1 <= Nat.max N1 N2)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_max_l.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with (max (max N1 N2) N3).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= Nat.max (Nat.max N1 N2) N3)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max (Nat.max N1 N2) N3 <=
 S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_max_l.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max (Nat.max N1 N2) N3 <=
 S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with (S (max (max N1 N2) N3)); apply le_n_Sn.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  assumption.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply Rle_lt_trans with
    (Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n)) +
      Rabs (Reste x y (pred n))).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) +
   Reste x y (Init.Nat.pred n)) <=
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n)) +
Rabs (Reste x y (Init.Nat.pred n))
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n)) +
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply Rabs_triang.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n)) +
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3 + eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply Rplus_lt_compat.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (sin x * sin y -
   B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n)) <
eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- Rabs_Ropp.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (-
   (sin x * sin y -
    B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n))) <
eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite Ropp_minus_distr.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs
  (B1 x (Init.Nat.pred n) * B1 y (Init.Nat.pred n) -
   sin x * sin y) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply H9.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Init.Nat.pred n >= N2)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold ge; apply le_trans with (max N1 N2).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N2 <= Nat.max N1 N2)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_max_r.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_S_n.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S (Nat.max N1 N2) <= S (Init.Nat.pred n))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- H12.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S (Nat.max N1 N2) <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S (Nat.max N1 N2) <= N)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S (Nat.max N1 N2) <=
 S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_n_S.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= S (Nat.max (Nat.max N1 N2) N3))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with (max (max N1 N2) N3).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max N1 N2 <= Nat.max (Nat.max N1 N2) N3)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max (Nat.max N1 N2) N3 <=
 S (Nat.max (Nat.max N1 N2) N3))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_max_l.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max (Nat.max N1 N2) N3 <=
 S (Nat.max (Nat.max N1 N2) N3))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_n_Sn.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  assumption.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n)) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  replace (Reste x y (pred n)) with (Reste x y (pred n) - 0).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Rabs (Reste x y (Init.Nat.pred n) - 0) < eps / 3
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply H10.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Init.Nat.pred n >= N3)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold ge.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N3 <= Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_S_n.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S N3 <= S (Init.Nat.pred n))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- H12.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S N3 <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S N3 <= N)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S N3 <= S (S (Nat.max (Nat.max N1 N2) N3)))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_n_S.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N3 <= S (Nat.max (Nat.max N1 N2) N3))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_trans with (max (max N1 N2) N3).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N3 <= Nat.max (Nat.max N1 N2) N3)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max (Nat.max N1 N2) N3 <=
 S (Nat.max (Nat.max N1 N2) N3))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_max_r.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Nat.max (Nat.max N1 N2) N3 <=
 S (Nat.max (Nat.max N1 N2) N3))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_n_Sn.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  assumption.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
Reste x y (Init.Nat.pred n) - 0 =
Reste x y (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  ring.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  pattern eps at 4; replace eps with (3 * (eps / 3)).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
eps / 3 + (eps / 3 + eps / 3) = 3 * (eps / 3)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
3 * (eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  ring.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
3 * (eps / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold Rdiv.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
3 * (eps * / 3) = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- Rmult_assoc.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
3 * eps * / 3 = eps
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply Rinv_r_simpl_m.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
3 <> 0
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  discrR.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply lt_le_trans with (pred N).
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(0 < Init.Nat.pred N)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Init.Nat.pred N <= Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold N; simpl; apply lt_O_Sn.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(Init.Nat.pred N <= Init.Nat.pred n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  apply le_S_n.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S (Init.Nat.pred N) <= S (Init.Nat.pred n))%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  rewrite <- H12.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(S (Init.Nat.pred N) <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  replace (S (pred N)) with N.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
(N <= n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
N = S (Init.Nat.pred N)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  assumption.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:n = S (Init.Nat.pred n)
N = S (Init.Nat.pred N)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  unfold N; simpl; reflexivity.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
n = S (Init.Nat.pred n)
  cut (0 < N)%nat.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat -> n = S (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat
  intro.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:(0 < N)%nat
n = S (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat
  cut (0 < n)%nat.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:(0 < N)%nat
(0 < n)%nat -> n = S (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:(0 < N)%nat
(0 < n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat
  intro.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:(0 < N)%nat
H13:(0 < n)%nat
n = S (Init.Nat.pred n)
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:(0 < N)%nat
(0 < n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat
  apply S_pred with 0%nat; assumption.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
H12:(0 < N)%nat
(0 < n)%nat
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat
  apply lt_le_trans with N; assumption.
x, y, eps:R
H:eps > 0
H0:Un_cv (A1 x) (cos x)
H1:Un_cv (A1 y) (cos y)
H2:Un_cv (B1 x) (sin x)
H3:Un_cv (B1 y) (sin y)
H4:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (A1 x n0 * A1 y n0 - cos x * cos y) < eps0
H5:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (B1 x n0 * B1 y n0 - sin x * sin y) < eps0
H6:forall eps0 : R,
eps0 > 0 ->
exists N0 : nat,
  forall n0 : nat,
  (n0 >= N0)%nat ->
  Rabs (Reste x y n0 - 0) < eps0
H7:0 < eps / 3
N1:nat
H8:forall n0 : nat,
(n0 >= N1)%nat ->
Rabs (A1 x n0 * A1 y n0 - cos x * cos y) <
eps / 3
N2:nat
H9:forall n0 : nat,
(n0 >= N2)%nat ->
Rabs (B1 x n0 * B1 y n0 - sin x * sin y) <
eps / 3
N3:nat
H10:forall n0 : nat, (n0 >= N3)%nat -> Rabs (Reste x y n0 - 0) < eps / 3
N:=S (S (Nat.max (Nat.max N1 N2) N3)):nat
n:nat
H11:(n >= N)%nat
(0 < N)%nat
  unfold N; apply lt_O_Sn.
Qed.